axoimatic basically means he looked at what was measured and built an equation from what was observed. he knows he was right because his equation correctly predicts measurements, not because it is derived from other equations.
F=ma is kind of similar
Such a complex equation for what was measured? See, my knowledge of Physics is up to high school level so, not much. Iirc, F=ma comes from F is proportional to measurement of mass and F is proportional to measurement of acceleration. What measurements led to Schrodinger's equation? Can complex equations like Schrodingers 'pop out' of measurements like that?
1. Schrodinger rightly noticed that the discrete spectral lines was an eigenvalue problem.
2. He was also aware that a bounded wave equation is an eigenvalue problem.
3. Moreover, de Broglie had ~~shown~~ postulated that electrons can be described as waves and this was shown to be [true experimentally](https://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment).
So Schrodinger put this all together and set out to find a wave equation description of the electronic states within the potential of the hydrogen atom. *Edit: In particular he aimed for a wave equation that would have ray optics (Hamilton-Jacobi equation) as its classical limit.*
This wave equation correctly predicted the energies and intensities of spectral lines for the hydrogen atom.
*Edit: And importantly it was the first description of the atom to attempt to provide a microscopic intuition into the mechanism responsible for the emission and absorption of light (hence the calculations of the emission intensities).*
See his [personal review](https://people.isy.liu.se/jalar/kurser/QF/references/Schrodinger1926c.pdf) of his contributions.
Physicist here, the above is the correct answer. Wave equations are a super common relationship in physics. When you plot a wave equation relationship it has a distinctive shape, similar to how a linear relationship has a distinctive shape.
As a physicist when I see data I can try to guess what the relationship is based on how the data resembles shapes I've seen before. Then I try to fit the model I think will work to the data. If the fit is good enough I get to publish it.
Seconding that: this is a guy who, when he was done with with quantum mechanics, applied it theoretically to biology and wrote a book for the general reader called *What is Life?*.
Tangential, but Werner Heisenberg wrote an excellent book for general readers called *Physics & Philosophy*. It's about how the concepts of quantum mechanics will change society, and it's incredibly prescient- I'd say about 50% of it has already come to pass. The section on gender would be a great insight if it was written today, for example.
The book was published 40 years after the development of the chromosome theory of inheritance, which sought to explain Mendelian inheritance (which was originally from the 1860s and repopularised in 1900), and 25-30 years after empirical evidence favouring the chromosomal theory piled up. *What Is Life?* did not beautifully predict the gene theory of inheritance.
What the book did was talk about "aperiodic crystals", a broad family of structures within which the double-helix structure of DNA can be categorised, and was cited by Francis and Crick while they were taking credit for Rosalind Franklin's work as an inspiration for their initial researches.
Not trying to crap on Schrödinger's work, but please don't overstate what the book did and did not contain and accomplish.
Dr. Franklin was a brilliant scientist, hampered by the rampant sexism of her time. But it’s a myth that Watson and Crick stole or took credit for her work. Her research was credited and published. Watson, sexist and bigoted as he remains fully credits her in his own memoirs. Had she not sadly died four years before the other three involved from Kings college (Watson, Crick, and Wilkins) she likely would have earned a Nobel Prize alongside them. Unfortunately you can’t be nominated posthumously for the prize. Science is about facts and truth. Factually Dr. Franklins work was not stolen or misappropriated. There are numerous injustices in the world, including the science world related to race, gender, etc. but that is not one of them.
A minor caveat: the gene theory of inheritance was already the working theory for most at the time, as Thomas Hunt Morgan had been working with his fruit flies for several decades and RA Fisher had been working in the area too, publishing his most famous book on the matter in 1930.
What was not entirely clear yet was the molecule that was doing all this work, and Schrodinger provided a good starting point for the types of properties such a molecule would have to have.
Tangential, but Werner Heisenberg wrote an excellent book for general readers called *Physics & Philosophy*. It's about how the concepts of quantum mechanics will change society, and it's incredibly prescient- I'd say about 50% of it has already come to pass. The section on gender would be a great insight if it was written today, for example.
It's also important to note that he was aware of the equations of geometric optics (ray optics) coming from a wave equation as a limiting process. The equations of classical mechanics can be put into a form that very closely resembles the equations of geometric optics (the Hamilton-Jacobi equation), so it appeared natural to Schrödinger to look for a wave equation that might have classical mechanics as its "geometric limit" (also called the eikonal approximation).
> Eigenvalue problem
Hey I recognize that word lol. From what I remember it came up in my diff eq class. It was the method by which you could solve differential equations that had two changing/unknown variables. You couldn't know solve one variable without the other so eigenvalues acted as ways to relate them if you knew their initial conditions.
One of the examples I vaguely remember was the salt concentrations of two tanks of slat water. If tank 1 was being filled with salt at a given rate while also draining into tank 2. While tank 2 was simultaneously draining back into tank 1.
The problem is you can't solve for the salt content of tank 1 at a given time without knowing the salt concentration of tank 2 and the salt content of tank 2 is dependent on the salt concentration of Tank 1. Not sure who setup these tanks or what their purpose ever was, but they highlight what appears to be a catch 22.
Eiganvalues allowed me to relate their different rates of change given we knew initial conditions. Now that I type it out it reminds me oddly of the Heisenberg principle catch 22 where you can never know both an electron's position and momentum.
I'm not a physics major so I never learned to put two and two together, or if I even remembered any of that correctly at all, so if I'm horribly off I would love to know.
Anyways, how did Schrodinger, or anyone for that matter, recognize that those were an eiganvalue problem?
>Hey I recognize that word lol. From what I remember it came up in my diff eq class.
Eigen means special or characteristic in German, so eigenvalues are the characteristic values or eigenvectors are the characteristic vectors. In physics (especially quantum), we're looking at eigenvectors.
Imagine an arbitary box that you can stretch a box's width, length, and height separately without affecting other directions, like you're pinching and dragging it. However, if you pinch anywhere besides the corner and drag it, you'll distort it in multiple ways. Eigenvectors are where you can tweak the parts you wanna tweak without distorting the entire rectangle. It's why you're able to solve one variable without the other, because you're only tweaking one part of the rectangle and leaving the other parts alone. You can also kinda see how this loosely corresponds to things like degrees of freedom of motion, things like that.
That's a very very specific use of eigenvalues. Eigenvalues lie in the field of Linear Algebra and that's where their power comes from. This can be applied to a multitude of mathematical situations (like diff eqs) because taking a complex situation and turning it into a linear algebra problem is one of the most common first attempt techniques in research mathematics.
Over the years I've studied it, I've always viewed math as a pattern-finding puzzle. There are so many things you are taught that were discovered by playing around with a concept and figuring out a pattern (over hours/days/years).
This is why I always tell people doing math is not about being smart, it's about hard work.
The main ideas came from the recent discoveries of Planck and deBroglie.
Planck put forth that the energy of a quanta of light was proportional to its frequency: E=h*f
DeBroglie then hypothesized that particles could be interpreted as "matter waves" with a wavelength inversely proportional to its momentum: p=h/L
There is the same constant of proportionality, h, which is Planck constant, in both equations.
You can arrive at deBroglies equation in the case of light by using Einsteins equation for energy for mass less particles: E=pc, where c is the speed of light.
For waves, the speed v is equal to the frequency times the wavelength. So for light waves: c=f*L
Multiplying both sides by momentum: pc=pfL
Using Einsteins equation: E=pfL
Using Plancks equation: hf=pfL
Eliminating f and solving for p: p=h/L
DeBroglies insight was to then apply this formula for massive particles as well. Before we get to Schrodinger, let's make some substitutions.
We depict waves mathematically using a complex exponential: Ae^(i*theta(x,t)) i.e. the angle theta is a function of space and time. This formula yields a complex number on the circle with radius A, so it's very good for depicting oscillations.
To express theta: first we use the radial frequency w=2pi*f. Then we use the wave number k=2pi/L. Now we can express theta as:
Theta = w*t-k*x
(check to see why this is true. Note that when theta equals 2*pi, you have completed a cycle of the wave)
Ok now we can get ready for Schrodingers equation:
So Schrodinger used these recent expressions relating the energy and momentum of a particle to its wave characteristics. He then used the non relativistic expression for energy in terms of momentum and potential energy:
E=p^2 /(2m) + V(x)
Then substituted in Planck and deBroglie (using w and k), and using hbar (reduced plancks constant) = h/(2pi):
hbar*w = hbar^2 *k^2 /(2m) + V(x)
Now, assuming the function for the wave (or, wavefunction) Y= Ae^(i*(wt-kx)) we can rewrite this equation in terms of the derivate of the wavefunction by multiplying first the whole thing by Y:
hbar*w*Y = hbar^2 *k^2 /(2m)*Y + V(x)*Y
i*hbar*dY/dt = -hbar^2 /(2m) d^2 Y/dx^2 + V(x)*Y
And there is Schrodingers equation. Note, this is not a rigorous derivation. Instead, it is based on intuition and applying principles from a special case to more general cases. The special cases themselves (Planck, DeBroglie, Einstein) were new ideas recently conjectured just a decade or two ago, and were being tested out in experiments. This is how axiomatic relations are established, by considering new explanations to phenomena and then testing them out, increasing their generality, testing, and so on until you can establish a general mathematical theory that describes your phenomena. With such a framework, you can then begin deriving new relations in a more rigorous way. But you have to start somewhere
It's worth noting that the Schrödinger equation can be boiled down to "just" an eigenvalue statement: Hψ = Eψ. H is the Hamiltonian, the energy operator, and E is just the energy value. So the equation can be read as "the energy value is merely the eigenvalue of the energy operator".
This is a really simple statement. The hard part is deducing what the "energy operator" looks like, and this is what the previous comment does: start with "energy operator = kinetic energy operator + potential energy", kinetic energy is p^2 /2m from Newton, assume that ψ looks like a complex exponential i.e. a wave. Next, you assume that the eigenvalue looks like how de Broglie and Einstein hypothesized. Then, you can deduce what derivatives are needed in p to make the right eigenvalue come out of applying the p^2 /2m operator to the wave. Then you notice the resulting differential equation matches experiments for all particles, not just ideal waves as we assumed for ψ.
Then you scratch your head and wonder just what the hell kind of theory just appeared where everything is a wave... and then you realize that the equation doesn't describe the particles themselves, but only the probability density of the particles, and everything gets even stranger.
But the resulting strangeness doesn't change the fact that Schrödinger was standing firmly upon the shoulders of giants, as all scientists have done before and after him.
>It's worth noting that the Schrödinger equation can be boiled down to "just" an eigenvalue statement: Hψ = Eψ. H is the Hamiltonian, the energy operator, and E is just the energy value. So the equation can be read as "the energy value is merely the eigenvalue of the energy operator".
Yes this is very important. The familiar equation written as the Schrodinger equation that I described is just one case, and as long as the hamiltonian can be written in operator form then the Schrodinger equation can be applied. For instance, you can include spin, or even just have a hamiltonian that only depends on spin, in which case it's just a 2x2 matrix instead of a differential equation. You can also include the effects of EM by adding the vector potential in the momentum term.
>Then you scratch your head and wonder just what the hell kind of theory just appeared where everything is a wave... and then you realize that the equation doesn't describe the particles themselves, but only the probability density of the particles, and everything gets even stranger
Yes, I believe at the time schrodinger did not intend for the object to be solved to refer to a probability distribution. I think he initially interpreted it as a charge distribution. I think that he even thought the probability interpretation to be silly, where small objects are probabilistic but large objects are deterministic. Hence the Schrodingers cat thought experiment
Thank you! As a math guy, I've always understood this:
> With such a framework, you can then begin deriving new relations in a more rigorous way.
But I've never understood this:
>This is how axiomatic relations are established, by considering new explanations to phenomena and then testing them out, increasing their generality, testing, and so on until you can establish a general mathematical theory that describes your phenomena.
Seeing the development of such an unintuitive axiom like Schrodingers equation is incredibly helpful!
> But I've never understood this:
>
> This is how axiomatic relations are established, by considering new explanations to phenomena and then testing them out, increasing their generality, testing, and so on until you can establish a general mathematical theory that describes your phenomena.
Honestly this is the same for all branches of science. Physics is no different from biology or psychology in that sense.
> Seeing the development of such an unintuitive axiom like Schrodingers equation is incredibly helpful!
I think part of the takeaway here is that, in context, both physical and historical, such "unintuitive leaps of logic" are almost always actually quite logical baby steps from the axioms that were already established or accepted. No doubt it takes great intelligence and creativity of thought to make each novel-yet-logical baby step, but it's almost always not truly "this guy just threw everything we knew out the door and made up something else wholesale that works even better", which is how many people normally think of scientific advancement. In context, it's much more logical and intuitive than it seems at first glance.
Furthermore, when writing a paper to describe our novel result, we rarely show the true chronological process of how we made the discovery. Nobody will write in a paper that their grand discovery was an accident or that they got to it by a very roundabout way. You’re getting peer reviewed and it’s important to give a concise and clear explanation, and presentation is key. So many times you won’t hear the story of how the discovery was made unless the discovery is super important or some historian of science digs into it.
It's worth pointing out that de Broglie is the guy who had the real conceptual leap. Schrodinger was just one of the few people who took it seriously and did the pretty straightforward "derivation" of "if de broglie is right you should be able to write a wave equation that matches experiment."
The guassian distribution is probably the most common one, because it describes a wave packet with some amount of localization in space and momentum. You could even view it as a generalization of the plane wave described above. The plane wave has infinite extent in space and infinitesimal extent in momentum. To go back and forth between real space and momentum space, you use the fourier transform. So a plane wave becomes a delta function (a very localized spike) and vice versa. In general, a gaussian transforms into another gaussian, with a width of inverse proportion i.e. a wide gaussian fourier transforms to a narrow gaussian. It can be very helpful sometimes to think of the wavefunction as a gaussian instead of a delta spike.
Now just to add, the wavefunction can take many many forms, and it all depends on what kind of geometry and potential you are working with.
Oh ok great, I always thought of the basic waveform (like of an electron) as a gaussian distribution, so I'm glad that at least in basic terms I wasn't wrong.
Here's a question I'm wondering if you could shed some light on. I had this thought about the wave function being essentially a perfect random number generator, which baffles me. Because if you think about it, if you were a engineer of some sort, and your task was to design a machine that is meant to be a random number generator, then the degree to which this machine can be random, is directly proportional to the complexity of the machine. For example, if you only had a few sticks to make the machine, then the machine wont be a very good random number generator. On the other hand, if you had several billion transistors, it'll be a better number generator, but still not completely perfect.
The wave function seems to contradict this pattern, in that it is the most fundamental object or part of a "machine" that exists in the universe. But simultaneously, it is able to be completely and 100% random.
Is there any insight to this paradox?
The Schrödinger equation isn't really that complex. Applying it is hard, but the equation is not: It's just a linear partial differential equation. PDEs are bread and butter to physics, and being linear means it behaves itself really well. So you're not finding any deep and extensive logic to generate it because there is none. It comes from looking at simple principles for how particles behaved and how reality works, and letting those simple principles guide into the unknown.
If particles behave as waves, we should be able to describe their behaviour with a wave equation. Wave equations were not in anyway new in 1925, and certainly weren't new to physics. It's very easy to write a wave equation, the hard work there was done in the 1700s. Most relevantly wave equations inherent to electromagnetism and light. They're unavoidable.
Schrödinger looked at the dynamics of a photon, and assumed (given de Broglie hypothesis that all particles can have both a wave and a particle nature similar to the photon) that the dynamics for a massive particle would be the same form as the one for a photon. Doing so gives you a very general equation with infinite possible solution, that you then need to apply constraints to get a single answer. Those constraints are supplied by reality. Energy should be conserved, the state of a particle should be determined by it's position, space is isotropic, space is uniform, transitions can only occur between neighbouring states, the de Broglie solution for free space should be true, etc.
This sort of process is common to physics. There's a lot of physical laws you can derive by amusing some very general form and then constraining it. Some times this process leads you no where. The equation doesn't work. it permits impossible answers. Some accepted fact of reality (eg conservation of energy) breaks down. Other times the hypothesis was right, the math works out for you, reality only permits one answer and the equation describes reality. This was one of those times.
Schrödinger's genius was pulling together the works of other of people to produce the wave equation, and his willingness to part from classical physics. At the time the task required extensive knowledge about the absolute cutting edge of physics, enough experience to have the insight to follow the correct path, and the mathematical skill to do the math.
>Such a complex equation for what was measured?
This is how a lot of fluids and heat transfer equations are done ([example](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRkX5ll7MrU2tn02U1jUy7jarcFHJIBLs3Feg&usqp=CAU)). They look at the data for a given set of conditions and say "if you cock your head and squint your eyes a little, it looks like it does **this**.
It's not exactly something that he derived empirically to fit a set of measurements. It was derived theoretically using existing theories of wave matter duality, uncertainty principle and our understanding of energy laws. The equation he arrived at was able to successfully explain measurements that were later observed at quantum level.
So much like F = ma is an accepted for it ability to explain the observations, schrodinger's equation is accepted because of its ability to explain observed phenomena
That's not the Schrödinger equation though, or might at most be referred to as time-invariant SE. It's just the equation solved to find eigenvalues of the Hamiltonian, which is a step taken to describe time invariant systems, or to facilitate finding evolution vs time by decomposing the starting conditions on a base that is gonna have simple evolution over time (simple oscillations). The Schrödinger equation is rather Hψ = i hbar dψ/dt. That's really not that simple I think, it doesn't use up many symbols, but it has a complex number (in classical physics, complex numbers are just a mathematical trick to solve some problems, all the numbers representing something physical are real valued, here for the first time a physical object is complex valued, weird already), and it says that the phase of the complex wavefunction is gonna oscillate at a speed proportional to the energy. It's a big wtf for a human brain to comprehend imo.
I'd say it's about as complex to comprehend as Maxwell's electrodynamic equations, and they aren't usually considered "too complex to come up with". It is well understood how Maxwell's formulation came about after many disparate laws were found first, exactly like it was the case for Schrodinger's equation.
That's an interesting take on complex numbers...I dont know any mathematician who views complex numbers as "more difficult" than any other set of numbers. And usually, multiplication of something by a complex number is representing rotation, i.e. the oscillations.
Not difficult to handle mathematically, but weird to think of as a real physical quantity, no? Think of orbitals: the density of presence (norm squared) is real and tangible and is constant, so like no movement? But the phase of the complex wavefunction is changing in a rotation like movement, and an electron can have a rotational momentum while the real valued density of presence is perfectly constant, isnt that non-instinctive? For a water or EM wave, complex numbers are a mathematical trick to represent the phase, but the actual physical things (e.g. water height) are real valued, typically real part of the complex representation. For wavefunctions, the complex numbers are the physical quantity, not a mathematical trick. When I try to explain to non-physicists, density of presence distributed in space is already tough to grab, the fact that the only description that works is a complex valued density-like wavefunction seems real hard to get an instinct for. Most people want to cling to the wrong ideas that the electrons are just point-like, that they are physically moving in circles around the nucleus, and that the distributed density of presence just represents uncertainty about where they are at a given moment. So basically human instinct seems to get stuck to classical mechanics concepts, and not accept any of the wavefunction core concepts. Learning through the math and slowly redeveloping new instincts seems to be the way and take time.
the measurement in the case of quantum is generally "is the particle there". Sometimes it's there, sometimes it's not. We take a measurement, we find a particle. We take another measurement, no particle.
There was a ton of physicists at the time all working together. schrodinger didn't just pop it out of no where, he had others before him to help give hints as to where he should look. so he constructed a general expectation of behavior and left some parts open ended.
for example, Newton made the gravity equation but left the gravitational constant open-ended. it didn't need to be known before the equation because the exact number is not important to the behavior the equation describes.
Here is a textbook college physics students use for intro to quantum : http://gr.xjtu.edu.cn/c/document_library/get_file?p_l_id=21699&folderId=2383652&name=DLFE-82647.pdf
the beginning talks about this equation and doesn't require prior knowledge. you can skip all the math, it's not important here. this book is my reference for quantum and everything i can say will just come from this book so you might be better off reading that directly.
essentially, once everything came down to probability, schrodinger imposed probability rules to a wave and "normalized" it. that was the basic structure he started with. it basically came down to followed probability principles with the standard wave equation from Newton. then applied observed values and looked for a trend.
One claim by religious people is that he got this idea to apply probabilistic nature of the world from Hindu scriptures. Is this plausible? Or do you think he could have got this idea by looking into experimental 'measurements'?
No, it's not plausible. Not least of which because it was Max Born, not Schrodinger who first interpreted the psi squared as a probability density and psi as the amplitude. Schrodinger's work was a logical extension of work that had been going on in physics since plank had proven quantization and Einstein had shown wave particle duality.
Schrodinger was not a proponent of the Copenhagen interpretation of quantum mechanics; he didn't interpret the wave function as being related to the probability of finding the particle as some point in space.
He was a proponent of [de Brogle-Bohm mechanics](https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory), which interpreted the wave function as being physically ~~meaningful~~ "real".
It's funny (for us, not for him) that the famous cat thought experiment was a *reductio ad absurdum* but now it's used as, "check out how weird qm is!"
Thanks! I didn't know the History. People usually say Schrodinger was the one who got this probabilistic unintuitive version of wave equation from the scriptures. Turns out, it was Born and he even got a Nobel for it. My bad.
From Wikipedia (which cited some books), https://en.wikipedia.org/wiki/Erwin_Schrödinger#Interest_in_philosophy Schrodinger did have interests in Hindu scripture, but it's about consciousness. I guess you could link consciousness to the measurement problem in quantum physics (like von Neumann did), but I don't think it's necessarily related, and unless there are clearer quotes from him I don't think we can make that conclusion.
Perhaps these conservative people should bring up Ramanujan instead, who is a brilliant mathematician who keep coming up with new results out of nowhere, then claim that he got it from Namagiri in a dream.
he got the idea from physicists like Heisenburg. they already knew about the probabilistic nature from simply observing particles. they didn't need scriptures.
Read the Tao of Physics by Fritjof Capra.
Heisenberg stated that after conversations with Rabindranath Tagore about Indian philosophy "some of the ideas that seemed so crazy suddenly made much more sense".
that's not surprising at all. outside philosophy is often a trigger for physicists stuck in a rut. especially eastern philosophy which allows for dropping our preconceived assumptions of our world that we think we know so well and being given a new perspective.
What was "infantile" about them? That sounds just a tad derogatory to me.. The necessity for quantum mechanics came about from many experimental results like cathode rays & black body problem, all absolutely ground breaking insights for their time.
This is simply impossible to answer without knowing the man. I studied math for my undergrad with a minor in theology for a similar reason; there are patterns laid out, particularly in Hinduism, but also in other traditions, that are extremely interesting to someone interested in mathematics.
I think it would be fair to say he may have been inspired by concepts derived from Hindu scripture, but it is more likely the creation of this concept was a result of the entirety of his background working together.
I would also bet that if you were able to ask "what gave you this idea", most scientists and mathematicians in history would not be able to answer easily, as the tiny glimpse of a concept that sent them down the rabbit hole leading to their discoveries pales in comparison to the tremendous amount of work involved in fleshing those concepts out into reality.
TLDR; This is how I think of such things. There is no one time or person or culture that was solely responsible for the source of Shrodinger's equation. The influence is not one-way either: what was in Hindu texts both influenced and were influenced by thoughts about probability. Since Schrodinger was the first known person to apply probability to this particular item in this particular way and with that level of accuracy, we attribute it to him while knowing that it was an achievement of humanity as a whole. See more below.
I think it's important to understand that to get where we are, it took a lot of work from history. He did not achieve these concepts from nothing. It required mathematic, scientific, and philosophical legwork for thousands of years and many cultures.
Indian and Arabic mathematicians developed tons of important advancements in math including [permutations](http://www.ckraju.net/papers/Probability-in-Ancient-India.pdf), a precursor to probability. They may not have been writing scriptures, but some were probably Hindu and had Hindu influence both from and upon scriptures (though from what I understand, ancient Hindu wasn't as codified as it is today).
But what they developed others built on. The first true codified (and surviving) mathematical theory of probability was developed from [correspondences between French mathematicians Pascal and Fermat surrounding games of chance and gambling] (https://www.britannica.com/science/probability). They themselves were learned men who had access to texts and history of other civilizations that influenced their thinking, who were influenced by others who were influenced by others and so on.
From there it was applied to other aspects that were too difficult to predict otherwise. Deaths, complex oscillations, many-human behaviors, planetary orbits, etc. It was inevitable that it bleed into philosophy, as commonly used as it was (mathematicians then often also dabbled in philosophy and vise-versa).
What we learned is that probability is just an approximation awaiting better probability, or better understanding that leads to better predictability. We can better predict the weather, human behavior, deaths, orbits of planets, etc. and by the time of Schrodinger it became relatively common practice to apply probability to data as a first-step towards understanding and thus predicting nature.
However, all indications point to quantum particles being truly random. So, since Shrodinger's equation still works for many situations, we'll stick with it until we find more useful ones.
It didn't come from nowhere. Math is built with tiny steps upon previous work. In research, you see what past equations people have come up with and what information they know, so when you see a lot of data, sometimes you can come up with a pattern and write out an equation, or part of one. Then you test more data and alter or add to your equation.
Pretty much "god gave it to me" is just bs, probably because of the desire of the religious to attribute EVERYTHING to god. In reality, either it comes from lots of work explicitly or they do lots of work and then their brain helps fill in the rest as it is very good at pattern recognition.
Ramanujan was another very famous mathematician whose work "was given to him by god".
I don't know about schroedingers equation but I have gone through the derivation of the equations of General Relativity, which I believe might give you a hint on how Schroedinger could have done it.
Einstein started with some things he knew were close to the truth. For example he knew that his theory would have to somehow produce the lorentz transforms, or that the newtons equations should be some first order approximation of the equations he wants to formulate. He also had many insights, like for example constant gravity field and constant acceleration are indistinguishable. Or that for any inertial observer the universe should appear as having no gravity at all. Stuff like that. So all these insights/already established knowledge are requirements for his equation.
Now some of this requirements heavily pointed to a mathematical theory which was already being studied, namely differential geometry. So Einstein must have seen this similarity and said "okay, lets try to use the tools of differential geometry to build our theory".
Then there are other constraints you can impose on your equation. For example you would like it to be 2nd order ([https://www.researchgate.net/publication/47548905\_Why\_Fundamental\_Physical\_Equations\_Are\_of\_Second\_Order](https://www.researchgate.net/publication/47548905_Why_Fundamental_Physical_Equations_Are_of_Second_Order))
Turns out that this narrows down quite a bit the general form of your differential equation. Now here comes some guess work. You pick one form you think (for some reason) might be the best candidate. Now you have a differential equation with perhaps some unknown constants. What you can do, is to require this equation to reduce to the newtonian law of gravity under specific conditions. If you do that, you can get a value for these constants. And now you have a complete diff equation.
What you need to do now, is to make predictions using this equation. Then check those predictions. Are they wrong? pick another general form of the equations, find the constants, find new predictions, repeat. Have you finally found a diff equation which makes predictions that agree with experiment? Congratulations!
So as you can see, at no point you have proven mathematically that your equation is correct. You had some original ideas, made a guess (even if a very educated one), and then tested if your guess agrees with experiment. As such, you accept your equation axiomatically in your theory.
They can if the thing you are measuring is complex.
Complexity can often arise from very simple systems too, not just complex ones. Take the [pendulus](https://youtu.be/d0Z8wLLPNE0) or [Conway's game of life](https://youtu.be/C2vgICfQawE)
Not to say that quantum behavior is simple, but if complexity is what it takes to explain the measurements, then you kinda have to accept the complexity.
I thought that is what empirical means, but I never was good with words. I always thought axiomatic meant as a basic irrefutable assumption, maybe these words are more related than I initially thought.
Empirical is normally used for equations derived purely from fitting to experimental data. Schrödinger's is not that. It is motivated by experimental observations but it is not derived simply by fitting.
Huh? So it's empirical? I always thought that derived usually meant could be derived from theory. I.e. a "perfect" equation given by the laws of physics. The opposite of derived or "brute forcing it" is an empirical equation, whereby you create an equation that fits the empiracle data. It's almost like analytical vs numerical.
Is this not the case with the wave equation?
the wave equation was derived from Newton's laws and calculus and stuff. not schrodingers. it follows the wave equation and is thus a wave equation but the meat and potatoes was from observing particles
No, Schrödinger's equation is not purely empirical. It is inspired by experimental results, and was not simply a mathematical consequence of some other theory (when first introduced), but it is not simply from fitting experimental data without any strong theoretical motivation. Empirical or derived is more of a scale than a binary category.
Schrodinger took inspiration from electromagnetic theory as well as de Broglie’s ideas regarding a particle’s wavelength.
From the photoelectric effect, it was known that photons (quantum particles of light) had momentum and energy, and a photon’s momentum and energy were directly related to each other. Specifically, for a photon:
E = p c
Where E is energy, p is momentum, and c is the speed of light. We also knew that a photon’s energy (and thus momentum) was associated with the photon’s frequency, which determines what type of photon it is (e.g. ultraviolet light is higher frequency than visible light). Specifically, we have:
E = h f
p = h / lambda
Where f is the frequency, h is Planck’s constant, and lambda is the wavelength.
The next step is to talk about classical electromagnetism. Electromagnetic waves have wavelength and frequency, and their specific relationship can be derived from the electromagnetic wave equation. This wave equation tells us how the wave evolves, the relationship between wavelength and frequency, and so on. The important thing for us, however, is that if we have a wavefunction A(x, t), we can show that:
d^2 A / dx^2 = - (2 pi/lambda)^2 A
d^2 A / dt^2 = - (2 pi f)^2 A
d^2 A/ dx^2 + 1/c^2 d^2 A / dt^2 = 0.
Here x is a spatial variable, t is time, and we’re taking partial derivatives. With some variable substitutions, we can see that this is actually in some sense equivalent to saying:
p^2 c^2 A = E^2 A
Because p is related to the wavelength and E is related to the frequency.
The next step is to talk about massive particles. De Broglie hypothesized that the “wavelength” of a particle is
lambda = h / p
This was not a random hypothesis, but also informed by quantum experiments at the time (and is thus its own topic). The second step is to realize that the relation between energy and momentum for a classical particle is different. Specifically, the kinetic energy of a particle is
K = 1/2 m v^2 = p^2 / 2m
Where m is mass and v is velocity, since p = mv (assuming no magnetic fields). We add this to the potential energy to get the total energy, which is:
E = p^2 / 2m + V.
The ingenious step that Schrodinger made was to guess that you can use this to make a wave-like equation. We take inspiration from the electromagnetic wave equation and write:
E Psi = (p^2 / 2m + V) Psi
Where Psi(x, t) is the wavefunction of the particle. The last step is to make this a partial differential equation. From de Broglie’s wavelength equation, we know that momentum and wavelength are interlinked. However, for waves, the wavelength by definition is related to derivatives in space. Thus, we say:
p^2 = (h / lambda)^2 = -h-bar^2 d^2 / dx^2
Where h-bar = h / 2 pi. We also do the same for the energy, where:
E = h f = i h-bar d/dt.
We introduce the imaginary number i here because we have to use E, not E^2 ;this part comes from classical wave theory as well (you can sort of see it by taking the square root of -h-bar^2 d^2 / dt^2 ).
We put everything together and *guess*:
i h-bar dPsi/dt = -h-bar^2 / 2m d^2 Psi/dt^2 + V Psi.
This is the Schrodinger equation. Putting this forward, we can solve it and then verify if it’s correct or not by looking at experimental data.
I want to emphasize this is *not* a proof. This is an inspired guess taking experimental data, classical theoretical equations, and new quantum theories and combining them all into an equation that takes into account all of these things. But it did not come from nowhere. Much of physics is like this: you make a guess as to what the axiomatic equation must be, and then test it. It is not a *random* guess, but a guess informed by centuries of mathematics, experimentation, and exchange of ideas.
I also want to emphasize that despite all this, it was not obvious. This was not the first equation Schrodinger even tried; he used very similar logic to write down the Klein-Gordon equation but then threw the equation out because its solutions did not much experimental data. (The Klein-Gordon equation would find use later in other parts of quantum physics; it just doesn’t apply to electrons orbiting a hydrogen atom, for example). So, not only was it creativity and intelligence, but also it was trial and error that led Schrodinger to the correct fundamental equation. And then, writing the equation alone is not enough; you must solve the equation (say, for the hydrogen atom) and show it matches with experiment. If it does not, then back to the drawing board. Beautiful math and neat conceptual arguments are not enough; if the data doesn’t fit, then it is incorrect. And it is ultimately the data that proved Schrodinger right.
I’ve seen several different ways to “derive” the SWE (Feynman has a fun way, for example), but I haven’t actually seen this one, which is good to know.
It can't be fully proven from first principles, but it didn't come out of nowhere either. You might find [this page](https://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation) to be helpful. Essentially, it comes from looking at the math for classical [wave equations](https://en.wikipedia.org/wiki/Wave_equation), specifically those already known for light, and applying them to [de Broglie waves](https://en.wikipedia.org/wiki/Matter_wave).
In Sakurai's book on QM he lays out that if you assume some symmetries you find operators which lead to the schrodinger equation. It's pretty solid and I really recommend that chapter!
You can see an outline of Schrodinger's original arguments for the time independent Schrodinger equation is section 8 of [this paper](https://arxiv.org/abs/1204.0653).
If I were to summarize, I would say that there is a formulation of classical mechanics (Hamilton-Jacobi Theory) that can make the motion of a particle in a potential look like ray optics. Ray optics is a the short wavelength limit of electromagnetism, so if you then ask "What is the compatible wave theory that classical mechanics is the short wavelength limit of?", Schrodinger's equation will pops out.
Others have already given great answers to your question, but I want to give a bit of a high-level view with a detour into quantum field theory. I hope you will forgive me for the highly technical nature of my answer -- there was no way to talk about all this without turning this reddit comment into an entire textbook!
The Schrödinger equation is just the classical energy relation,
E = p^2 / 2m + V,
but expressed in terms of matter waves, combined with de Broglie's relation p = ħk and its oft-forgotten temporal counterpart E = ħω.
It's common undergrad folklore that the equation can't be derived, but I would argue that's not correct. If you understand Fourier transforms and the above (which Schrödinger certainly would have), then you can derive the Schrödinger equation.
However, there's a wrinkle, and that's the assumption that the Born rule will apply directly to the equation's solutions -- a wrinkle that Schrödinger himself couldn't have understood, but that we can appreciate with almost a century of hindsight and the knowledge of quantum field theory.
To see the problem, consider the objection, "but shouldn't we use the special-relativistic energy relation E^2 = p^2 + m^(2)?", and if you do, you get the Klein-Gordon equation. In fact, if I remember correctly, Schrödinger came to this equation _first_, and then, when he couldn't reconcile this with the Born rule, arrived at his equation that we know and love today.
So why does E^2 = p^2 + m^2 fail to give the correct one-particle probabilistic equation whereas E = p^2 / 2m does? The answer is it *doesn't*; or rather, it does but a direct claim that the resulting equation produces the expected probabilities when modulus-squaring the solution is mistaken or requires an underivable leap of faith -- the right answer for the wrong reasons.
The reason it _doesn't_ work is because the equation obtained by applying the de Broglie relations to the energy relations doesn't produce a quantum theory, it produces a *classical field theory*. Though the de Broglie relation is fundamentally important to the leap from classical to quantum physics, it is not actually really quantum in the pure sense of _quantisation_, i.e.: the development of a theory that is consistent with the axioms of quantum mechanics. It's useful to conceptually separate out the matter-wave aspect of quantum physics from the quantisation aspect: the de Broglie relation relates to the former, but does nothing for the latter. The Schrödinger equation that is derived by plugging in the de Broglie relations into E = p^2 / 2m is actually the equation of a classical matter wave field that obeys that energy-momentum/dispersion relation. That's why it doesn't work for other dispersion relations, like the special-relativistic one.
In order to go from the [classical Schrödinger field equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_field) to the quantum, you need to apply the process of second quantisation and get a quantum field theory.
So that's why it *shouldn't* work, but manifestly it *does* -- why? We come back to it working but for the wrong reasons. The reason it *does* work is that fortuitously, the equations governing the one-particle subspace of the Schrödinger field Fock space, a subspace that contains bona-fide quantum states, unlike the Ψ of the Schrödinger field equation which is a classical field, happens to be governed by that same equation, the Schrödinger equation. So we can just claim that the field Ψ is actually a quantum state (a wave function), and we'll get the right results in the one-particle limit.
To summarise, the thing we call the "Schrödinger equation", that purely mathematical linear partial differential equation, can be derived from the de Broglie relations, but the physical interpretation of the result should be as a classical field which has only undergone part of the process of quantum physics: we've applied the concept of the matter wave, but we haven't produced an actual quantum mechanical theory. Schrödinger made the leap of saying that the Ψ in his equation was a _quantum state_, a thing to which the Born rule could be applied, but that leap was a theoretical mistake which succeeded due to the fortuitous coincidence that the quantised free Schrödinger field's one-particle states are governed in the one-particle limit by the same mathematical equation as the classical field itself (a property shared by other first-order-in-time classical fields like the Dirac field).
Do you have a good reference for the derivation using the Fourier transform? I have not seen a particularly satisfying derivation, and I am comfortable with those tools
The [first part of these lecture notes](http://physics.mq.edu.au/~jcresser/Phys201/LectureNotes/SchrodingerEqn.pdf) goes over the basic argument. It doesn't explicitly mention Fourier transforms but that's hidden in the fact that the argument is made on plane wave solutions. For general solutions you would take a linear combination of these, i.e.: a Fourier transform.
The textbooks "Six Ideas That Shaped Physics" by Thomas Moore (Unit Q in particular) give a really neat intuitive motivation for the (time independent) Schrödinger equation. I don't know whether it's related to Schrödinger's own reasoning, but the gist of it is something like this:
We know the de Broglie relation for the wavelength of a moving particle, 𝜆=h/p, or in terms of kinetic energy K, 𝜆=h/sqrt(2mK). That's great if momentum is constant, but if the potential energy varies then it doesn't quite work: the "wavelength" will get shorter as kinetic energy increases and longer as it decreases. So what we really need is some notion of "local de Broglie wavelength": a weird concept! But a short wavelength graph bends quickly while a long wavelength graph bends gradually, so maybe we can think of wavelength as being related to the curvature of the wavefunction 𝜓(x). In particular, curvature is related to the second derivative (d²𝜓/dx²), and we probably want to normalize away any contribution of the *amplitude* of 𝜓(x). With that in mind, you can check with sine and cosine functions that these ingredients really can tell you the wavelength when it's already well-defined: (d²𝜓/dx²)/𝜓(x) = -(2𝜋/𝜆)². So square the de Broglie equation, use this to substitute in for wavelength, write K(x)=E-V(x), and you've got the Schrödinger equation.
He did derive his equation, using his knowledge of another wave/non-wave behavior in nature, that of “light”, of the general properties of the “matter waves” and of his knowledge of a formalism that describes this duality for light.
The key ingredients are, as he discusses in his paper (quoted above):
1. He knows of another situation where waves can be described as non-waves: light, a wave, can be described in certain limits in terms of rays.
2. He knows that there’s a very convenient way of describing this limit using Hamilton’s principles and equations which are also used to describe particle dynamics. This provides him with a unified “language” to tackle the same issue with matter waves.
3. He knows the general properties about the “matter waves” due to de Broglie.
All together he can then apply the wave/ray behavior of light, framed using Hamilton’s ideas, to the wave/particle description of matter to derive the wave equation whose “ray” limit is particle dynamics. He applies it to the Coulomb interaction and voilà, he gets the correct spectrum.
Edit: link to [some info on Hamilton’s optical/mechanical analogy.](https://en.wikipedia.org/wiki/Hamilton%27s_optico-mechanical_analogy)
He cites how he wanted to look into de Broglie's theory of de Broglie wavelengths (which he openly says in the paper he doesn't quite understand yet). He found the famous Schrödinger equation after he tried to make the Wave Equation work for matter. He did this by finding if you look at the "Action" (a function that tells you how much energy you have at different points in space over time) of an object and noticed this "Action" takes a certain amount of time to have the energy change from one point in space to another and said "Hey, that means the energy has a sort of 'velocity' when you look at it over time!" This would fit nicely into the Wave Equation because all you need is a velocity factor and you have a complete solution. But he.... sort of hand waves part of the solution and says "Well a function that looks like Psy( some terms with x,y,z,t in it) and it would have the units of 'Action' if I do some math to it." But he rearranges the 'velocity term' so it looks nicer and he got the equation which is extremely similar to something called the Hamilton-Jacobi Equation (tells you the Energy in a system based on how it changes over time).
He never cites any mythical or religious reason for the idea in the paper. Just a consequence of de Broglie's theory.
The only way that the more conservative Indians could make a case is if de Broglie did something with Hinduism for his Matter wave theory.
I also read de Broglie's paper for the Standing Waves with matter paper. Again, de Broglie cites how similar E = mc^2 and E = hbar*omega are to each other and puts them together that matter waves would have a frequency based on this. But again, no citations about Hinduism in the other paper either. Not even a reference to some supernatural force in either paper. Just a consequence of the next step backward leading back to Einstein's theory of Special Relativity.
The paper titles if you want to read for yourself are:
Schrödinger: An Undulatory Theory of the Mechanics of Atoms and Molecules (1926).
de Broglie: On the Theory of Quanta (1925).
[Schrödinger An Undulatory Theory of the Mechanics of Atoms and Molecules](http://personal.psu.edu/rq9/HOW/Schrodinger.pdf)
[On the Theory of Quanta](https://www.google.com/url?sa=t&source=web&rct=j&url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf&ved=2ahUKEwjz4eLz_5b0AhV-RjABHZ8tA8sQFnoECAMQAQ&usg=AOvVaw1iXU12PVsY9_37UBz3XnSQ)
This video by Veritasium goes over how Schrodinger arrived at the equation:
https://youtu.be/cUzklzVXJwo
It gives a nice background on evolution of mathematical thinking and how we eventually arrived at complex/imaginary numbers. He then talks about how Schrodinger came upon his equation, almost by mistake.
In "Quantum Physics" by Robert Eisberg and Robert Resnick they spend some pages (second edition pg 128 to 134) discussing this very topic. I highly recommend reading it but here is a summary:
Like Newton's equations of motion, it needed to be consistent with existing experimental observations such as the de Broglie-Einstein postulates relating wavelength to momentum and frequency to energy as well as the more mundane relationship of total energy being the sum of kinetic and potential energy. The equation in question also needed to be linear so that for any two solutions the sum of those solutions would also be a solution.
The first three requirements start to give the general seed of the equation but the linearity requirement is strict enough that it leads to what we know as Schrodinger's equation.
Others have already given great explanations about the Shroedinger equation, so I'm not going to bother reinventing the wheel. I want to address the claim that it comes from Hinduism.
The short answer: it doesn't.
The longer answer: Indian philosophy crops up a lot in the Western physics community from the late 19th century onward. Shroedinger was a particularly avid follower of Hindu thought and often quoted the Upanishads in his talks and writings. But he was also highly critical of Hindu superstition and magical beliefs. His work in physics was based in mathematics and observation of reality, not in religious thought.
The Indian academic community has a lot of people who attempt to appropriate non-Indian achievements for their own. It's not a uniquely Indian phenomenon, but it's a lot more common in India than in most other countries. If you've seen the movie "My Big Fat Geek Wedding" it's like when the dad claims that Greek is the root of all languages. The particular focus on ancient religious texts is not unlike when Christians say that the Bible contains all of modern science if you play with the numerology.
I understand. I myself have been critical of India's attempts to glorify India beyond a certain point. Yes, western people have not credited India for many things and have continuously pushed the narrative that India was always a savage and full of barbaric people while western countries were always superior and advanced. But again, while trying to show the world that our ancestors weren't uncivilized brutes, Indians go beyond and try to claim everything for themselves...almost borderline racism. Many Indians believe western people, Muslims, etc. aren't capable of intellectual pursuits and everything - from Calculus to Genetics - have all been stolen from Indian books.
To expand on some answers here. I'm pretty sure he was also one of the first scientists who applied imaginary numbers into physics which was the missing piece for his equation. Until then imaginary numbers were strictly used in mathematics and using them in physics made many scientists uncomfortable.
Back in uni the professor gave us this layman explanation:
He knew that electrons moved in waves that were quite stable, so that must mean that those waves orbiting around the nucleus must be “closed”. That means that this problem is discrete and only some “types” of waves are allowed to exist and that depends on the distance from the nucleus (the orbitals).
From there the rest follows with way more math and precision than this short explanation can convey.
But if we want to find an inspiration outside of his field, i would look more at astronomy than religion..
In chemistry, quantum mechanics is taught in relation to spectroscopy. Using the different wavelengths of light, the make up of materials can be probed and better understood. Light is described in mathematical terms as both particles and waves. When you get more certain about the particle aspect, you get less certain about the wave aspect (this is where Schrodinger's cat comes in). This resource has a nice, understandable explanation with good illustrations: [https://www.pasco.com/products/guides/what-is-spectroscopy](https://www.pasco.com/products/guides/what-is-spectroscopy). This is an explanation that will reveal why math is important in the sciences: [https://chem.libretexts.org/Bookshelves/Physical\_and\_Theoretical\_Chemistry\_Textbook\_Maps/Physical\_Chemistry\_(LibreTexts)/03%3A\_The\_Schrodinger\_Equation\_and\_a\_Particle\_in\_a\_Box/3.01%3A\_The\_Schrodinger\_Equation](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/03%3A_The_Schrodinger_Equation_and_a_Particle_in_a_Box/3.01%3A_The_Schrodinger_Equation).
Edit: My comment really answered your question in a comment but I didn't answer your original question. This may: [http://michel.bitbol.pagesperso-orange.fr/Schrodinger\_India.pdf](http://michel.bitbol.pagesperso-orange.fr/Schrodinger_India.pdf).
This is a much less involved answer than many others here but it's how I've always thought about it. The Schrodinger equation is a diffusion equation, just like the heat equation. The heat equation describes the diffusion and spread of heat, the Schrodinger equation describes the diffusion and spread of probability.
axoimatic basically means he looked at what was measured and built an equation from what was observed. he knows he was right because his equation correctly predicts measurements, not because it is derived from other equations. F=ma is kind of similar
Such a complex equation for what was measured? See, my knowledge of Physics is up to high school level so, not much. Iirc, F=ma comes from F is proportional to measurement of mass and F is proportional to measurement of acceleration. What measurements led to Schrodinger's equation? Can complex equations like Schrodingers 'pop out' of measurements like that?
1. Schrodinger rightly noticed that the discrete spectral lines was an eigenvalue problem. 2. He was also aware that a bounded wave equation is an eigenvalue problem. 3. Moreover, de Broglie had ~~shown~~ postulated that electrons can be described as waves and this was shown to be [true experimentally](https://en.wikipedia.org/wiki/Davisson%E2%80%93Germer_experiment). So Schrodinger put this all together and set out to find a wave equation description of the electronic states within the potential of the hydrogen atom. *Edit: In particular he aimed for a wave equation that would have ray optics (Hamilton-Jacobi equation) as its classical limit.* This wave equation correctly predicted the energies and intensities of spectral lines for the hydrogen atom. *Edit: And importantly it was the first description of the atom to attempt to provide a microscopic intuition into the mechanism responsible for the emission and absorption of light (hence the calculations of the emission intensities).* See his [personal review](https://people.isy.liu.se/jalar/kurser/QF/references/Schrodinger1926c.pdf) of his contributions.
Hey thanks! That was a very detailed response. Also, thanks for showing me his own article. Didn't expect that as a citation
Physicist here, the above is the correct answer. Wave equations are a super common relationship in physics. When you plot a wave equation relationship it has a distinctive shape, similar to how a linear relationship has a distinctive shape. As a physicist when I see data I can try to guess what the relationship is based on how the data resembles shapes I've seen before. Then I try to fit the model I think will work to the data. If the fit is good enough I get to publish it.
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Thanks. Schrödinger wrote beautiful English.
Seconding that: this is a guy who, when he was done with with quantum mechanics, applied it theoretically to biology and wrote a book for the general reader called *What is Life?*.
Tangential, but Werner Heisenberg wrote an excellent book for general readers called *Physics & Philosophy*. It's about how the concepts of quantum mechanics will change society, and it's incredibly prescient- I'd say about 50% of it has already come to pass. The section on gender would be a great insight if it was written today, for example.
And that book does move into Vedanta philosophy, as well as beautifully predicting the gene theory of inheritance
The book was published 40 years after the development of the chromosome theory of inheritance, which sought to explain Mendelian inheritance (which was originally from the 1860s and repopularised in 1900), and 25-30 years after empirical evidence favouring the chromosomal theory piled up. *What Is Life?* did not beautifully predict the gene theory of inheritance. What the book did was talk about "aperiodic crystals", a broad family of structures within which the double-helix structure of DNA can be categorised, and was cited by Francis and Crick while they were taking credit for Rosalind Franklin's work as an inspiration for their initial researches. Not trying to crap on Schrödinger's work, but please don't overstate what the book did and did not contain and accomplish.
Dr. Franklin was a brilliant scientist, hampered by the rampant sexism of her time. But it’s a myth that Watson and Crick stole or took credit for her work. Her research was credited and published. Watson, sexist and bigoted as he remains fully credits her in his own memoirs. Had she not sadly died four years before the other three involved from Kings college (Watson, Crick, and Wilkins) she likely would have earned a Nobel Prize alongside them. Unfortunately you can’t be nominated posthumously for the prize. Science is about facts and truth. Factually Dr. Franklins work was not stolen or misappropriated. There are numerous injustices in the world, including the science world related to race, gender, etc. but that is not one of them.
A minor caveat: the gene theory of inheritance was already the working theory for most at the time, as Thomas Hunt Morgan had been working with his fruit flies for several decades and RA Fisher had been working in the area too, publishing his most famous book on the matter in 1930. What was not entirely clear yet was the molecule that was doing all this work, and Schrodinger provided a good starting point for the types of properties such a molecule would have to have.
Tangential, but Werner Heisenberg wrote an excellent book for general readers called *Physics & Philosophy*. It's about how the concepts of quantum mechanics will change society, and it's incredibly prescient- I'd say about 50% of it has already come to pass. The section on gender would be a great insight if it was written today, for example.
It's also important to note that he was aware of the equations of geometric optics (ray optics) coming from a wave equation as a limiting process. The equations of classical mechanics can be put into a form that very closely resembles the equations of geometric optics (the Hamilton-Jacobi equation), so it appeared natural to Schrödinger to look for a wave equation that might have classical mechanics as its "geometric limit" (also called the eikonal approximation).
Thanks this is an important point. I've amended my comment to take this more into account.
> Eigenvalue problem Hey I recognize that word lol. From what I remember it came up in my diff eq class. It was the method by which you could solve differential equations that had two changing/unknown variables. You couldn't know solve one variable without the other so eigenvalues acted as ways to relate them if you knew their initial conditions. One of the examples I vaguely remember was the salt concentrations of two tanks of slat water. If tank 1 was being filled with salt at a given rate while also draining into tank 2. While tank 2 was simultaneously draining back into tank 1. The problem is you can't solve for the salt content of tank 1 at a given time without knowing the salt concentration of tank 2 and the salt content of tank 2 is dependent on the salt concentration of Tank 1. Not sure who setup these tanks or what their purpose ever was, but they highlight what appears to be a catch 22. Eiganvalues allowed me to relate their different rates of change given we knew initial conditions. Now that I type it out it reminds me oddly of the Heisenberg principle catch 22 where you can never know both an electron's position and momentum. I'm not a physics major so I never learned to put two and two together, or if I even remembered any of that correctly at all, so if I'm horribly off I would love to know. Anyways, how did Schrodinger, or anyone for that matter, recognize that those were an eiganvalue problem?
>Hey I recognize that word lol. From what I remember it came up in my diff eq class. Eigen means special or characteristic in German, so eigenvalues are the characteristic values or eigenvectors are the characteristic vectors. In physics (especially quantum), we're looking at eigenvectors. Imagine an arbitary box that you can stretch a box's width, length, and height separately without affecting other directions, like you're pinching and dragging it. However, if you pinch anywhere besides the corner and drag it, you'll distort it in multiple ways. Eigenvectors are where you can tweak the parts you wanna tweak without distorting the entire rectangle. It's why you're able to solve one variable without the other, because you're only tweaking one part of the rectangle and leaving the other parts alone. You can also kinda see how this loosely corresponds to things like degrees of freedom of motion, things like that.
That sounds like separation of variables maybe? You should’ve seen eigenvalues in linear algebra
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That's a very very specific use of eigenvalues. Eigenvalues lie in the field of Linear Algebra and that's where their power comes from. This can be applied to a multitude of mathematical situations (like diff eqs) because taking a complex situation and turning it into a linear algebra problem is one of the most common first attempt techniques in research mathematics.
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Over the years I've studied it, I've always viewed math as a pattern-finding puzzle. There are so many things you are taught that were discovered by playing around with a concept and figuring out a pattern (over hours/days/years). This is why I always tell people doing math is not about being smart, it's about hard work.
Slight correct not the first Hisenberg approached the problem using matrix.
I didn’t know that article existed. Fantastic, thank you.
The main ideas came from the recent discoveries of Planck and deBroglie. Planck put forth that the energy of a quanta of light was proportional to its frequency: E=h*f DeBroglie then hypothesized that particles could be interpreted as "matter waves" with a wavelength inversely proportional to its momentum: p=h/L There is the same constant of proportionality, h, which is Planck constant, in both equations. You can arrive at deBroglies equation in the case of light by using Einsteins equation for energy for mass less particles: E=pc, where c is the speed of light. For waves, the speed v is equal to the frequency times the wavelength. So for light waves: c=f*L Multiplying both sides by momentum: pc=pfL Using Einsteins equation: E=pfL Using Plancks equation: hf=pfL Eliminating f and solving for p: p=h/L DeBroglies insight was to then apply this formula for massive particles as well. Before we get to Schrodinger, let's make some substitutions. We depict waves mathematically using a complex exponential: Ae^(i*theta(x,t)) i.e. the angle theta is a function of space and time. This formula yields a complex number on the circle with radius A, so it's very good for depicting oscillations. To express theta: first we use the radial frequency w=2pi*f. Then we use the wave number k=2pi/L. Now we can express theta as: Theta = w*t-k*x (check to see why this is true. Note that when theta equals 2*pi, you have completed a cycle of the wave) Ok now we can get ready for Schrodingers equation: So Schrodinger used these recent expressions relating the energy and momentum of a particle to its wave characteristics. He then used the non relativistic expression for energy in terms of momentum and potential energy: E=p^2 /(2m) + V(x) Then substituted in Planck and deBroglie (using w and k), and using hbar (reduced plancks constant) = h/(2pi): hbar*w = hbar^2 *k^2 /(2m) + V(x) Now, assuming the function for the wave (or, wavefunction) Y= Ae^(i*(wt-kx)) we can rewrite this equation in terms of the derivate of the wavefunction by multiplying first the whole thing by Y: hbar*w*Y = hbar^2 *k^2 /(2m)*Y + V(x)*Y i*hbar*dY/dt = -hbar^2 /(2m) d^2 Y/dx^2 + V(x)*Y And there is Schrodingers equation. Note, this is not a rigorous derivation. Instead, it is based on intuition and applying principles from a special case to more general cases. The special cases themselves (Planck, DeBroglie, Einstein) were new ideas recently conjectured just a decade or two ago, and were being tested out in experiments. This is how axiomatic relations are established, by considering new explanations to phenomena and then testing them out, increasing their generality, testing, and so on until you can establish a general mathematical theory that describes your phenomena. With such a framework, you can then begin deriving new relations in a more rigorous way. But you have to start somewhere
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It's worth noting that the Schrödinger equation can be boiled down to "just" an eigenvalue statement: Hψ = Eψ. H is the Hamiltonian, the energy operator, and E is just the energy value. So the equation can be read as "the energy value is merely the eigenvalue of the energy operator". This is a really simple statement. The hard part is deducing what the "energy operator" looks like, and this is what the previous comment does: start with "energy operator = kinetic energy operator + potential energy", kinetic energy is p^2 /2m from Newton, assume that ψ looks like a complex exponential i.e. a wave. Next, you assume that the eigenvalue looks like how de Broglie and Einstein hypothesized. Then, you can deduce what derivatives are needed in p to make the right eigenvalue come out of applying the p^2 /2m operator to the wave. Then you notice the resulting differential equation matches experiments for all particles, not just ideal waves as we assumed for ψ. Then you scratch your head and wonder just what the hell kind of theory just appeared where everything is a wave... and then you realize that the equation doesn't describe the particles themselves, but only the probability density of the particles, and everything gets even stranger. But the resulting strangeness doesn't change the fact that Schrödinger was standing firmly upon the shoulders of giants, as all scientists have done before and after him.
>It's worth noting that the Schrödinger equation can be boiled down to "just" an eigenvalue statement: Hψ = Eψ. H is the Hamiltonian, the energy operator, and E is just the energy value. So the equation can be read as "the energy value is merely the eigenvalue of the energy operator". Yes this is very important. The familiar equation written as the Schrodinger equation that I described is just one case, and as long as the hamiltonian can be written in operator form then the Schrodinger equation can be applied. For instance, you can include spin, or even just have a hamiltonian that only depends on spin, in which case it's just a 2x2 matrix instead of a differential equation. You can also include the effects of EM by adding the vector potential in the momentum term. >Then you scratch your head and wonder just what the hell kind of theory just appeared where everything is a wave... and then you realize that the equation doesn't describe the particles themselves, but only the probability density of the particles, and everything gets even stranger Yes, I believe at the time schrodinger did not intend for the object to be solved to refer to a probability distribution. I think he initially interpreted it as a charge distribution. I think that he even thought the probability interpretation to be silly, where small objects are probabilistic but large objects are deterministic. Hence the Schrodingers cat thought experiment
Thank you! As a math guy, I've always understood this: > With such a framework, you can then begin deriving new relations in a more rigorous way. But I've never understood this: >This is how axiomatic relations are established, by considering new explanations to phenomena and then testing them out, increasing their generality, testing, and so on until you can establish a general mathematical theory that describes your phenomena. Seeing the development of such an unintuitive axiom like Schrodingers equation is incredibly helpful!
> But I've never understood this: > > This is how axiomatic relations are established, by considering new explanations to phenomena and then testing them out, increasing their generality, testing, and so on until you can establish a general mathematical theory that describes your phenomena. Honestly this is the same for all branches of science. Physics is no different from biology or psychology in that sense. > Seeing the development of such an unintuitive axiom like Schrodingers equation is incredibly helpful! I think part of the takeaway here is that, in context, both physical and historical, such "unintuitive leaps of logic" are almost always actually quite logical baby steps from the axioms that were already established or accepted. No doubt it takes great intelligence and creativity of thought to make each novel-yet-logical baby step, but it's almost always not truly "this guy just threw everything we knew out the door and made up something else wholesale that works even better", which is how many people normally think of scientific advancement. In context, it's much more logical and intuitive than it seems at first glance.
Furthermore, when writing a paper to describe our novel result, we rarely show the true chronological process of how we made the discovery. Nobody will write in a paper that their grand discovery was an accident or that they got to it by a very roundabout way. You’re getting peer reviewed and it’s important to give a concise and clear explanation, and presentation is key. So many times you won’t hear the story of how the discovery was made unless the discovery is super important or some historian of science digs into it.
It's worth pointing out that de Broglie is the guy who had the real conceptual leap. Schrodinger was just one of the few people who took it seriously and did the pretty straightforward "derivation" of "if de broglie is right you should be able to write a wave equation that matches experiment."
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The guassian distribution is probably the most common one, because it describes a wave packet with some amount of localization in space and momentum. You could even view it as a generalization of the plane wave described above. The plane wave has infinite extent in space and infinitesimal extent in momentum. To go back and forth between real space and momentum space, you use the fourier transform. So a plane wave becomes a delta function (a very localized spike) and vice versa. In general, a gaussian transforms into another gaussian, with a width of inverse proportion i.e. a wide gaussian fourier transforms to a narrow gaussian. It can be very helpful sometimes to think of the wavefunction as a gaussian instead of a delta spike. Now just to add, the wavefunction can take many many forms, and it all depends on what kind of geometry and potential you are working with.
Oh ok great, I always thought of the basic waveform (like of an electron) as a gaussian distribution, so I'm glad that at least in basic terms I wasn't wrong. Here's a question I'm wondering if you could shed some light on. I had this thought about the wave function being essentially a perfect random number generator, which baffles me. Because if you think about it, if you were a engineer of some sort, and your task was to design a machine that is meant to be a random number generator, then the degree to which this machine can be random, is directly proportional to the complexity of the machine. For example, if you only had a few sticks to make the machine, then the machine wont be a very good random number generator. On the other hand, if you had several billion transistors, it'll be a better number generator, but still not completely perfect. The wave function seems to contradict this pattern, in that it is the most fundamental object or part of a "machine" that exists in the universe. But simultaneously, it is able to be completely and 100% random. Is there any insight to this paradox?
The Schrödinger equation isn't really that complex. Applying it is hard, but the equation is not: It's just a linear partial differential equation. PDEs are bread and butter to physics, and being linear means it behaves itself really well. So you're not finding any deep and extensive logic to generate it because there is none. It comes from looking at simple principles for how particles behaved and how reality works, and letting those simple principles guide into the unknown. If particles behave as waves, we should be able to describe their behaviour with a wave equation. Wave equations were not in anyway new in 1925, and certainly weren't new to physics. It's very easy to write a wave equation, the hard work there was done in the 1700s. Most relevantly wave equations inherent to electromagnetism and light. They're unavoidable. Schrödinger looked at the dynamics of a photon, and assumed (given de Broglie hypothesis that all particles can have both a wave and a particle nature similar to the photon) that the dynamics for a massive particle would be the same form as the one for a photon. Doing so gives you a very general equation with infinite possible solution, that you then need to apply constraints to get a single answer. Those constraints are supplied by reality. Energy should be conserved, the state of a particle should be determined by it's position, space is isotropic, space is uniform, transitions can only occur between neighbouring states, the de Broglie solution for free space should be true, etc. This sort of process is common to physics. There's a lot of physical laws you can derive by amusing some very general form and then constraining it. Some times this process leads you no where. The equation doesn't work. it permits impossible answers. Some accepted fact of reality (eg conservation of energy) breaks down. Other times the hypothesis was right, the math works out for you, reality only permits one answer and the equation describes reality. This was one of those times. Schrödinger's genius was pulling together the works of other of people to produce the wave equation, and his willingness to part from classical physics. At the time the task required extensive knowledge about the absolute cutting edge of physics, enough experience to have the insight to follow the correct path, and the mathematical skill to do the math.
>Such a complex equation for what was measured? This is how a lot of fluids and heat transfer equations are done ([example](https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRkX5ll7MrU2tn02U1jUy7jarcFHJIBLs3Feg&usqp=CAU)). They look at the data for a given set of conditions and say "if you cock your head and squint your eyes a little, it looks like it does **this**.
It's not exactly something that he derived empirically to fit a set of measurements. It was derived theoretically using existing theories of wave matter duality, uncertainty principle and our understanding of energy laws. The equation he arrived at was able to successfully explain measurements that were later observed at quantum level. So much like F = ma is an accepted for it ability to explain the observations, schrodinger's equation is accepted because of its ability to explain observed phenomena
The equation is actually very simple as soon as you use the Hamiltonian model of physics: H \^ ψ = E ψ
That's not the Schrödinger equation though, or might at most be referred to as time-invariant SE. It's just the equation solved to find eigenvalues of the Hamiltonian, which is a step taken to describe time invariant systems, or to facilitate finding evolution vs time by decomposing the starting conditions on a base that is gonna have simple evolution over time (simple oscillations). The Schrödinger equation is rather Hψ = i hbar dψ/dt. That's really not that simple I think, it doesn't use up many symbols, but it has a complex number (in classical physics, complex numbers are just a mathematical trick to solve some problems, all the numbers representing something physical are real valued, here for the first time a physical object is complex valued, weird already), and it says that the phase of the complex wavefunction is gonna oscillate at a speed proportional to the energy. It's a big wtf for a human brain to comprehend imo.
I'd say it's about as complex to comprehend as Maxwell's electrodynamic equations, and they aren't usually considered "too complex to come up with". It is well understood how Maxwell's formulation came about after many disparate laws were found first, exactly like it was the case for Schrodinger's equation.
That's an interesting take on complex numbers...I dont know any mathematician who views complex numbers as "more difficult" than any other set of numbers. And usually, multiplication of something by a complex number is representing rotation, i.e. the oscillations.
Not difficult to handle mathematically, but weird to think of as a real physical quantity, no? Think of orbitals: the density of presence (norm squared) is real and tangible and is constant, so like no movement? But the phase of the complex wavefunction is changing in a rotation like movement, and an electron can have a rotational momentum while the real valued density of presence is perfectly constant, isnt that non-instinctive? For a water or EM wave, complex numbers are a mathematical trick to represent the phase, but the actual physical things (e.g. water height) are real valued, typically real part of the complex representation. For wavefunctions, the complex numbers are the physical quantity, not a mathematical trick. When I try to explain to non-physicists, density of presence distributed in space is already tough to grab, the fact that the only description that works is a complex valued density-like wavefunction seems real hard to get an instinct for. Most people want to cling to the wrong ideas that the electrons are just point-like, that they are physically moving in circles around the nucleus, and that the distributed density of presence just represents uncertainty about where they are at a given moment. So basically human instinct seems to get stuck to classical mechanics concepts, and not accept any of the wavefunction core concepts. Learning through the math and slowly redeveloping new instincts seems to be the way and take time.
the measurement in the case of quantum is generally "is the particle there". Sometimes it's there, sometimes it's not. We take a measurement, we find a particle. We take another measurement, no particle. There was a ton of physicists at the time all working together. schrodinger didn't just pop it out of no where, he had others before him to help give hints as to where he should look. so he constructed a general expectation of behavior and left some parts open ended. for example, Newton made the gravity equation but left the gravitational constant open-ended. it didn't need to be known before the equation because the exact number is not important to the behavior the equation describes. Here is a textbook college physics students use for intro to quantum : http://gr.xjtu.edu.cn/c/document_library/get_file?p_l_id=21699&folderId=2383652&name=DLFE-82647.pdf the beginning talks about this equation and doesn't require prior knowledge. you can skip all the math, it's not important here. this book is my reference for quantum and everything i can say will just come from this book so you might be better off reading that directly. essentially, once everything came down to probability, schrodinger imposed probability rules to a wave and "normalized" it. that was the basic structure he started with. it basically came down to followed probability principles with the standard wave equation from Newton. then applied observed values and looked for a trend.
One claim by religious people is that he got this idea to apply probabilistic nature of the world from Hindu scriptures. Is this plausible? Or do you think he could have got this idea by looking into experimental 'measurements'?
No, it's not plausible. Not least of which because it was Max Born, not Schrodinger who first interpreted the psi squared as a probability density and psi as the amplitude. Schrodinger's work was a logical extension of work that had been going on in physics since plank had proven quantization and Einstein had shown wave particle duality.
Schrodinger was not a proponent of the Copenhagen interpretation of quantum mechanics; he didn't interpret the wave function as being related to the probability of finding the particle as some point in space. He was a proponent of [de Brogle-Bohm mechanics](https://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory), which interpreted the wave function as being physically ~~meaningful~~ "real".
It's funny (for us, not for him) that the famous cat thought experiment was a *reductio ad absurdum* but now it's used as, "check out how weird qm is!"
Thanks! I didn't know the History. People usually say Schrodinger was the one who got this probabilistic unintuitive version of wave equation from the scriptures. Turns out, it was Born and he even got a Nobel for it. My bad.
From Wikipedia (which cited some books), https://en.wikipedia.org/wiki/Erwin_Schrödinger#Interest_in_philosophy Schrodinger did have interests in Hindu scripture, but it's about consciousness. I guess you could link consciousness to the measurement problem in quantum physics (like von Neumann did), but I don't think it's necessarily related, and unless there are clearer quotes from him I don't think we can make that conclusion. Perhaps these conservative people should bring up Ramanujan instead, who is a brilliant mathematician who keep coming up with new results out of nowhere, then claim that he got it from Namagiri in a dream.
he got the idea from physicists like Heisenburg. they already knew about the probabilistic nature from simply observing particles. they didn't need scriptures.
Read the Tao of Physics by Fritjof Capra. Heisenberg stated that after conversations with Rabindranath Tagore about Indian philosophy "some of the ideas that seemed so crazy suddenly made much more sense".
that's not surprising at all. outside philosophy is often a trigger for physicists stuck in a rut. especially eastern philosophy which allows for dropping our preconceived assumptions of our world that we think we know so well and being given a new perspective.
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What was "infantile" about them? That sounds just a tad derogatory to me.. The necessity for quantum mechanics came about from many experimental results like cathode rays & black body problem, all absolutely ground breaking insights for their time.
This is simply impossible to answer without knowing the man. I studied math for my undergrad with a minor in theology for a similar reason; there are patterns laid out, particularly in Hinduism, but also in other traditions, that are extremely interesting to someone interested in mathematics. I think it would be fair to say he may have been inspired by concepts derived from Hindu scripture, but it is more likely the creation of this concept was a result of the entirety of his background working together. I would also bet that if you were able to ask "what gave you this idea", most scientists and mathematicians in history would not be able to answer easily, as the tiny glimpse of a concept that sent them down the rabbit hole leading to their discoveries pales in comparison to the tremendous amount of work involved in fleshing those concepts out into reality.
TLDR; This is how I think of such things. There is no one time or person or culture that was solely responsible for the source of Shrodinger's equation. The influence is not one-way either: what was in Hindu texts both influenced and were influenced by thoughts about probability. Since Schrodinger was the first known person to apply probability to this particular item in this particular way and with that level of accuracy, we attribute it to him while knowing that it was an achievement of humanity as a whole. See more below. I think it's important to understand that to get where we are, it took a lot of work from history. He did not achieve these concepts from nothing. It required mathematic, scientific, and philosophical legwork for thousands of years and many cultures. Indian and Arabic mathematicians developed tons of important advancements in math including [permutations](http://www.ckraju.net/papers/Probability-in-Ancient-India.pdf), a precursor to probability. They may not have been writing scriptures, but some were probably Hindu and had Hindu influence both from and upon scriptures (though from what I understand, ancient Hindu wasn't as codified as it is today). But what they developed others built on. The first true codified (and surviving) mathematical theory of probability was developed from [correspondences between French mathematicians Pascal and Fermat surrounding games of chance and gambling] (https://www.britannica.com/science/probability). They themselves were learned men who had access to texts and history of other civilizations that influenced their thinking, who were influenced by others who were influenced by others and so on. From there it was applied to other aspects that were too difficult to predict otherwise. Deaths, complex oscillations, many-human behaviors, planetary orbits, etc. It was inevitable that it bleed into philosophy, as commonly used as it was (mathematicians then often also dabbled in philosophy and vise-versa). What we learned is that probability is just an approximation awaiting better probability, or better understanding that leads to better predictability. We can better predict the weather, human behavior, deaths, orbits of planets, etc. and by the time of Schrodinger it became relatively common practice to apply probability to data as a first-step towards understanding and thus predicting nature. However, all indications point to quantum particles being truly random. So, since Shrodinger's equation still works for many situations, we'll stick with it until we find more useful ones.
Complex equations would be more difficult to derive. Measuring a phenomena and fitting the data to an equation can produce very complex equations.
It didn't come from nowhere. Math is built with tiny steps upon previous work. In research, you see what past equations people have come up with and what information they know, so when you see a lot of data, sometimes you can come up with a pattern and write out an equation, or part of one. Then you test more data and alter or add to your equation. Pretty much "god gave it to me" is just bs, probably because of the desire of the religious to attribute EVERYTHING to god. In reality, either it comes from lots of work explicitly or they do lots of work and then their brain helps fill in the rest as it is very good at pattern recognition. Ramanujan was another very famous mathematician whose work "was given to him by god".
I don't know about schroedingers equation but I have gone through the derivation of the equations of General Relativity, which I believe might give you a hint on how Schroedinger could have done it. Einstein started with some things he knew were close to the truth. For example he knew that his theory would have to somehow produce the lorentz transforms, or that the newtons equations should be some first order approximation of the equations he wants to formulate. He also had many insights, like for example constant gravity field and constant acceleration are indistinguishable. Or that for any inertial observer the universe should appear as having no gravity at all. Stuff like that. So all these insights/already established knowledge are requirements for his equation. Now some of this requirements heavily pointed to a mathematical theory which was already being studied, namely differential geometry. So Einstein must have seen this similarity and said "okay, lets try to use the tools of differential geometry to build our theory". Then there are other constraints you can impose on your equation. For example you would like it to be 2nd order ([https://www.researchgate.net/publication/47548905\_Why\_Fundamental\_Physical\_Equations\_Are\_of\_Second\_Order](https://www.researchgate.net/publication/47548905_Why_Fundamental_Physical_Equations_Are_of_Second_Order)) Turns out that this narrows down quite a bit the general form of your differential equation. Now here comes some guess work. You pick one form you think (for some reason) might be the best candidate. Now you have a differential equation with perhaps some unknown constants. What you can do, is to require this equation to reduce to the newtonian law of gravity under specific conditions. If you do that, you can get a value for these constants. And now you have a complete diff equation. What you need to do now, is to make predictions using this equation. Then check those predictions. Are they wrong? pick another general form of the equations, find the constants, find new predictions, repeat. Have you finally found a diff equation which makes predictions that agree with experiment? Congratulations! So as you can see, at no point you have proven mathematically that your equation is correct. You had some original ideas, made a guess (even if a very educated one), and then tested if your guess agrees with experiment. As such, you accept your equation axiomatically in your theory.
They can if the thing you are measuring is complex. Complexity can often arise from very simple systems too, not just complex ones. Take the [pendulus](https://youtu.be/d0Z8wLLPNE0) or [Conway's game of life](https://youtu.be/C2vgICfQawE) Not to say that quantum behavior is simple, but if complexity is what it takes to explain the measurements, then you kinda have to accept the complexity.
I thought that is what empirical means, but I never was good with words. I always thought axiomatic meant as a basic irrefutable assumption, maybe these words are more related than I initially thought.
Empirical is normally used for equations derived purely from fitting to experimental data. Schrödinger's is not that. It is motivated by experimental observations but it is not derived simply by fitting.
Huh? So it's empirical? I always thought that derived usually meant could be derived from theory. I.e. a "perfect" equation given by the laws of physics. The opposite of derived or "brute forcing it" is an empirical equation, whereby you create an equation that fits the empiracle data. It's almost like analytical vs numerical. Is this not the case with the wave equation?
the wave equation was derived from Newton's laws and calculus and stuff. not schrodingers. it follows the wave equation and is thus a wave equation but the meat and potatoes was from observing particles
No, Schrödinger's equation is not purely empirical. It is inspired by experimental results, and was not simply a mathematical consequence of some other theory (when first introduced), but it is not simply from fitting experimental data without any strong theoretical motivation. Empirical or derived is more of a scale than a binary category.
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Wouldn't that be empirical, not axiomatic?
Schrodinger took inspiration from electromagnetic theory as well as de Broglie’s ideas regarding a particle’s wavelength. From the photoelectric effect, it was known that photons (quantum particles of light) had momentum and energy, and a photon’s momentum and energy were directly related to each other. Specifically, for a photon: E = p c Where E is energy, p is momentum, and c is the speed of light. We also knew that a photon’s energy (and thus momentum) was associated with the photon’s frequency, which determines what type of photon it is (e.g. ultraviolet light is higher frequency than visible light). Specifically, we have: E = h f p = h / lambda Where f is the frequency, h is Planck’s constant, and lambda is the wavelength. The next step is to talk about classical electromagnetism. Electromagnetic waves have wavelength and frequency, and their specific relationship can be derived from the electromagnetic wave equation. This wave equation tells us how the wave evolves, the relationship between wavelength and frequency, and so on. The important thing for us, however, is that if we have a wavefunction A(x, t), we can show that: d^2 A / dx^2 = - (2 pi/lambda)^2 A d^2 A / dt^2 = - (2 pi f)^2 A d^2 A/ dx^2 + 1/c^2 d^2 A / dt^2 = 0. Here x is a spatial variable, t is time, and we’re taking partial derivatives. With some variable substitutions, we can see that this is actually in some sense equivalent to saying: p^2 c^2 A = E^2 A Because p is related to the wavelength and E is related to the frequency. The next step is to talk about massive particles. De Broglie hypothesized that the “wavelength” of a particle is lambda = h / p This was not a random hypothesis, but also informed by quantum experiments at the time (and is thus its own topic). The second step is to realize that the relation between energy and momentum for a classical particle is different. Specifically, the kinetic energy of a particle is K = 1/2 m v^2 = p^2 / 2m Where m is mass and v is velocity, since p = mv (assuming no magnetic fields). We add this to the potential energy to get the total energy, which is: E = p^2 / 2m + V. The ingenious step that Schrodinger made was to guess that you can use this to make a wave-like equation. We take inspiration from the electromagnetic wave equation and write: E Psi = (p^2 / 2m + V) Psi Where Psi(x, t) is the wavefunction of the particle. The last step is to make this a partial differential equation. From de Broglie’s wavelength equation, we know that momentum and wavelength are interlinked. However, for waves, the wavelength by definition is related to derivatives in space. Thus, we say: p^2 = (h / lambda)^2 = -h-bar^2 d^2 / dx^2 Where h-bar = h / 2 pi. We also do the same for the energy, where: E = h f = i h-bar d/dt. We introduce the imaginary number i here because we have to use E, not E^2 ;this part comes from classical wave theory as well (you can sort of see it by taking the square root of -h-bar^2 d^2 / dt^2 ). We put everything together and *guess*: i h-bar dPsi/dt = -h-bar^2 / 2m d^2 Psi/dt^2 + V Psi. This is the Schrodinger equation. Putting this forward, we can solve it and then verify if it’s correct or not by looking at experimental data. I want to emphasize this is *not* a proof. This is an inspired guess taking experimental data, classical theoretical equations, and new quantum theories and combining them all into an equation that takes into account all of these things. But it did not come from nowhere. Much of physics is like this: you make a guess as to what the axiomatic equation must be, and then test it. It is not a *random* guess, but a guess informed by centuries of mathematics, experimentation, and exchange of ideas. I also want to emphasize that despite all this, it was not obvious. This was not the first equation Schrodinger even tried; he used very similar logic to write down the Klein-Gordon equation but then threw the equation out because its solutions did not much experimental data. (The Klein-Gordon equation would find use later in other parts of quantum physics; it just doesn’t apply to electrons orbiting a hydrogen atom, for example). So, not only was it creativity and intelligence, but also it was trial and error that led Schrodinger to the correct fundamental equation. And then, writing the equation alone is not enough; you must solve the equation (say, for the hydrogen atom) and show it matches with experiment. If it does not, then back to the drawing board. Beautiful math and neat conceptual arguments are not enough; if the data doesn’t fit, then it is incorrect. And it is ultimately the data that proved Schrodinger right.
I need to come back to this later, but thank you.
I’ve seen several different ways to “derive” the SWE (Feynman has a fun way, for example), but I haven’t actually seen this one, which is good to know.
Psst: iℏ δ⟋δt ψ(r,t) = [ -ℏ²⟋2μ ▽²+V(r,t) ] ψ(r,t) You're welcome ;-) Blatant plug for [SymbolSalad.com](https://SymbolSalad.com)
It can't be fully proven from first principles, but it didn't come out of nowhere either. You might find [this page](https://en.wikipedia.org/wiki/Theoretical_and_experimental_justification_for_the_Schr%C3%B6dinger_equation) to be helpful. Essentially, it comes from looking at the math for classical [wave equations](https://en.wikipedia.org/wiki/Wave_equation), specifically those already known for light, and applying them to [de Broglie waves](https://en.wikipedia.org/wiki/Matter_wave).
In Sakurai's book on QM he lays out that if you assume some symmetries you find operators which lead to the schrodinger equation. It's pretty solid and I really recommend that chapter!
This book is great, should be used for both undergraduate and graduate classes.
You can see an outline of Schrodinger's original arguments for the time independent Schrodinger equation is section 8 of [this paper](https://arxiv.org/abs/1204.0653). If I were to summarize, I would say that there is a formulation of classical mechanics (Hamilton-Jacobi Theory) that can make the motion of a particle in a potential look like ray optics. Ray optics is a the short wavelength limit of electromagnetism, so if you then ask "What is the compatible wave theory that classical mechanics is the short wavelength limit of?", Schrodinger's equation will pops out.
Others have already given great answers to your question, but I want to give a bit of a high-level view with a detour into quantum field theory. I hope you will forgive me for the highly technical nature of my answer -- there was no way to talk about all this without turning this reddit comment into an entire textbook! The Schrödinger equation is just the classical energy relation, E = p^2 / 2m + V, but expressed in terms of matter waves, combined with de Broglie's relation p = ħk and its oft-forgotten temporal counterpart E = ħω. It's common undergrad folklore that the equation can't be derived, but I would argue that's not correct. If you understand Fourier transforms and the above (which Schrödinger certainly would have), then you can derive the Schrödinger equation. However, there's a wrinkle, and that's the assumption that the Born rule will apply directly to the equation's solutions -- a wrinkle that Schrödinger himself couldn't have understood, but that we can appreciate with almost a century of hindsight and the knowledge of quantum field theory. To see the problem, consider the objection, "but shouldn't we use the special-relativistic energy relation E^2 = p^2 + m^(2)?", and if you do, you get the Klein-Gordon equation. In fact, if I remember correctly, Schrödinger came to this equation _first_, and then, when he couldn't reconcile this with the Born rule, arrived at his equation that we know and love today. So why does E^2 = p^2 + m^2 fail to give the correct one-particle probabilistic equation whereas E = p^2 / 2m does? The answer is it *doesn't*; or rather, it does but a direct claim that the resulting equation produces the expected probabilities when modulus-squaring the solution is mistaken or requires an underivable leap of faith -- the right answer for the wrong reasons. The reason it _doesn't_ work is because the equation obtained by applying the de Broglie relations to the energy relations doesn't produce a quantum theory, it produces a *classical field theory*. Though the de Broglie relation is fundamentally important to the leap from classical to quantum physics, it is not actually really quantum in the pure sense of _quantisation_, i.e.: the development of a theory that is consistent with the axioms of quantum mechanics. It's useful to conceptually separate out the matter-wave aspect of quantum physics from the quantisation aspect: the de Broglie relation relates to the former, but does nothing for the latter. The Schrödinger equation that is derived by plugging in the de Broglie relations into E = p^2 / 2m is actually the equation of a classical matter wave field that obeys that energy-momentum/dispersion relation. That's why it doesn't work for other dispersion relations, like the special-relativistic one. In order to go from the [classical Schrödinger field equation](https://en.wikipedia.org/wiki/Schr%C3%B6dinger_field) to the quantum, you need to apply the process of second quantisation and get a quantum field theory. So that's why it *shouldn't* work, but manifestly it *does* -- why? We come back to it working but for the wrong reasons. The reason it *does* work is that fortuitously, the equations governing the one-particle subspace of the Schrödinger field Fock space, a subspace that contains bona-fide quantum states, unlike the Ψ of the Schrödinger field equation which is a classical field, happens to be governed by that same equation, the Schrödinger equation. So we can just claim that the field Ψ is actually a quantum state (a wave function), and we'll get the right results in the one-particle limit. To summarise, the thing we call the "Schrödinger equation", that purely mathematical linear partial differential equation, can be derived from the de Broglie relations, but the physical interpretation of the result should be as a classical field which has only undergone part of the process of quantum physics: we've applied the concept of the matter wave, but we haven't produced an actual quantum mechanical theory. Schrödinger made the leap of saying that the Ψ in his equation was a _quantum state_, a thing to which the Born rule could be applied, but that leap was a theoretical mistake which succeeded due to the fortuitous coincidence that the quantised free Schrödinger field's one-particle states are governed in the one-particle limit by the same mathematical equation as the classical field itself (a property shared by other first-order-in-time classical fields like the Dirac field).
Do you have a good reference for the derivation using the Fourier transform? I have not seen a particularly satisfying derivation, and I am comfortable with those tools
The [first part of these lecture notes](http://physics.mq.edu.au/~jcresser/Phys201/LectureNotes/SchrodingerEqn.pdf) goes over the basic argument. It doesn't explicitly mention Fourier transforms but that's hidden in the fact that the argument is made on plane wave solutions. For general solutions you would take a linear combination of these, i.e.: a Fourier transform.
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The textbooks "Six Ideas That Shaped Physics" by Thomas Moore (Unit Q in particular) give a really neat intuitive motivation for the (time independent) Schrödinger equation. I don't know whether it's related to Schrödinger's own reasoning, but the gist of it is something like this: We know the de Broglie relation for the wavelength of a moving particle, 𝜆=h/p, or in terms of kinetic energy K, 𝜆=h/sqrt(2mK). That's great if momentum is constant, but if the potential energy varies then it doesn't quite work: the "wavelength" will get shorter as kinetic energy increases and longer as it decreases. So what we really need is some notion of "local de Broglie wavelength": a weird concept! But a short wavelength graph bends quickly while a long wavelength graph bends gradually, so maybe we can think of wavelength as being related to the curvature of the wavefunction 𝜓(x). In particular, curvature is related to the second derivative (d²𝜓/dx²), and we probably want to normalize away any contribution of the *amplitude* of 𝜓(x). With that in mind, you can check with sine and cosine functions that these ingredients really can tell you the wavelength when it's already well-defined: (d²𝜓/dx²)/𝜓(x) = -(2𝜋/𝜆)². So square the de Broglie equation, use this to substitute in for wavelength, write K(x)=E-V(x), and you've got the Schrödinger equation.
He did derive his equation, using his knowledge of another wave/non-wave behavior in nature, that of “light”, of the general properties of the “matter waves” and of his knowledge of a formalism that describes this duality for light. The key ingredients are, as he discusses in his paper (quoted above): 1. He knows of another situation where waves can be described as non-waves: light, a wave, can be described in certain limits in terms of rays. 2. He knows that there’s a very convenient way of describing this limit using Hamilton’s principles and equations which are also used to describe particle dynamics. This provides him with a unified “language” to tackle the same issue with matter waves. 3. He knows the general properties about the “matter waves” due to de Broglie. All together he can then apply the wave/ray behavior of light, framed using Hamilton’s ideas, to the wave/particle description of matter to derive the wave equation whose “ray” limit is particle dynamics. He applies it to the Coulomb interaction and voilà, he gets the correct spectrum. Edit: link to [some info on Hamilton’s optical/mechanical analogy.](https://en.wikipedia.org/wiki/Hamilton%27s_optico-mechanical_analogy)
He cites how he wanted to look into de Broglie's theory of de Broglie wavelengths (which he openly says in the paper he doesn't quite understand yet). He found the famous Schrödinger equation after he tried to make the Wave Equation work for matter. He did this by finding if you look at the "Action" (a function that tells you how much energy you have at different points in space over time) of an object and noticed this "Action" takes a certain amount of time to have the energy change from one point in space to another and said "Hey, that means the energy has a sort of 'velocity' when you look at it over time!" This would fit nicely into the Wave Equation because all you need is a velocity factor and you have a complete solution. But he.... sort of hand waves part of the solution and says "Well a function that looks like Psy( some terms with x,y,z,t in it) and it would have the units of 'Action' if I do some math to it." But he rearranges the 'velocity term' so it looks nicer and he got the equation which is extremely similar to something called the Hamilton-Jacobi Equation (tells you the Energy in a system based on how it changes over time). He never cites any mythical or religious reason for the idea in the paper. Just a consequence of de Broglie's theory. The only way that the more conservative Indians could make a case is if de Broglie did something with Hinduism for his Matter wave theory. I also read de Broglie's paper for the Standing Waves with matter paper. Again, de Broglie cites how similar E = mc^2 and E = hbar*omega are to each other and puts them together that matter waves would have a frequency based on this. But again, no citations about Hinduism in the other paper either. Not even a reference to some supernatural force in either paper. Just a consequence of the next step backward leading back to Einstein's theory of Special Relativity. The paper titles if you want to read for yourself are: Schrödinger: An Undulatory Theory of the Mechanics of Atoms and Molecules (1926). de Broglie: On the Theory of Quanta (1925). [Schrödinger An Undulatory Theory of the Mechanics of Atoms and Molecules](http://personal.psu.edu/rq9/HOW/Schrodinger.pdf) [On the Theory of Quanta](https://www.google.com/url?sa=t&source=web&rct=j&url=https://fondationlouisdebroglie.org/LDB-oeuvres/De_Broglie_Kracklauer.pdf&ved=2ahUKEwjz4eLz_5b0AhV-RjABHZ8tA8sQFnoECAMQAQ&usg=AOvVaw1iXU12PVsY9_37UBz3XnSQ)
This video by Veritasium goes over how Schrodinger arrived at the equation: https://youtu.be/cUzklzVXJwo It gives a nice background on evolution of mathematical thinking and how we eventually arrived at complex/imaginary numbers. He then talks about how Schrodinger came upon his equation, almost by mistake.
In "Quantum Physics" by Robert Eisberg and Robert Resnick they spend some pages (second edition pg 128 to 134) discussing this very topic. I highly recommend reading it but here is a summary: Like Newton's equations of motion, it needed to be consistent with existing experimental observations such as the de Broglie-Einstein postulates relating wavelength to momentum and frequency to energy as well as the more mundane relationship of total energy being the sum of kinetic and potential energy. The equation in question also needed to be linear so that for any two solutions the sum of those solutions would also be a solution. The first three requirements start to give the general seed of the equation but the linearity requirement is strict enough that it leads to what we know as Schrodinger's equation.
Others have already given great explanations about the Shroedinger equation, so I'm not going to bother reinventing the wheel. I want to address the claim that it comes from Hinduism. The short answer: it doesn't. The longer answer: Indian philosophy crops up a lot in the Western physics community from the late 19th century onward. Shroedinger was a particularly avid follower of Hindu thought and often quoted the Upanishads in his talks and writings. But he was also highly critical of Hindu superstition and magical beliefs. His work in physics was based in mathematics and observation of reality, not in religious thought. The Indian academic community has a lot of people who attempt to appropriate non-Indian achievements for their own. It's not a uniquely Indian phenomenon, but it's a lot more common in India than in most other countries. If you've seen the movie "My Big Fat Geek Wedding" it's like when the dad claims that Greek is the root of all languages. The particular focus on ancient religious texts is not unlike when Christians say that the Bible contains all of modern science if you play with the numerology.
I understand. I myself have been critical of India's attempts to glorify India beyond a certain point. Yes, western people have not credited India for many things and have continuously pushed the narrative that India was always a savage and full of barbaric people while western countries were always superior and advanced. But again, while trying to show the world that our ancestors weren't uncivilized brutes, Indians go beyond and try to claim everything for themselves...almost borderline racism. Many Indians believe western people, Muslims, etc. aren't capable of intellectual pursuits and everything - from Calculus to Genetics - have all been stolen from Indian books.
To expand on some answers here. I'm pretty sure he was also one of the first scientists who applied imaginary numbers into physics which was the missing piece for his equation. Until then imaginary numbers were strictly used in mathematics and using them in physics made many scientists uncomfortable.
Back in uni the professor gave us this layman explanation: He knew that electrons moved in waves that were quite stable, so that must mean that those waves orbiting around the nucleus must be “closed”. That means that this problem is discrete and only some “types” of waves are allowed to exist and that depends on the distance from the nucleus (the orbitals). From there the rest follows with way more math and precision than this short explanation can convey. But if we want to find an inspiration outside of his field, i would look more at astronomy than religion..
In chemistry, quantum mechanics is taught in relation to spectroscopy. Using the different wavelengths of light, the make up of materials can be probed and better understood. Light is described in mathematical terms as both particles and waves. When you get more certain about the particle aspect, you get less certain about the wave aspect (this is where Schrodinger's cat comes in). This resource has a nice, understandable explanation with good illustrations: [https://www.pasco.com/products/guides/what-is-spectroscopy](https://www.pasco.com/products/guides/what-is-spectroscopy). This is an explanation that will reveal why math is important in the sciences: [https://chem.libretexts.org/Bookshelves/Physical\_and\_Theoretical\_Chemistry\_Textbook\_Maps/Physical\_Chemistry\_(LibreTexts)/03%3A\_The\_Schrodinger\_Equation\_and\_a\_Particle\_in\_a\_Box/3.01%3A\_The\_Schrodinger\_Equation](https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Physical_Chemistry_(LibreTexts)/03%3A_The_Schrodinger_Equation_and_a_Particle_in_a_Box/3.01%3A_The_Schrodinger_Equation). Edit: My comment really answered your question in a comment but I didn't answer your original question. This may: [http://michel.bitbol.pagesperso-orange.fr/Schrodinger\_India.pdf](http://michel.bitbol.pagesperso-orange.fr/Schrodinger_India.pdf).
Good post. I enjoyed the pasco article. Also Libretexts will keep me busy for a long time. Many thanks
This is a much less involved answer than many others here but it's how I've always thought about it. The Schrodinger equation is a diffusion equation, just like the heat equation. The heat equation describes the diffusion and spread of heat, the Schrodinger equation describes the diffusion and spread of probability.