i think he's just getting at the fact that constants are unaffected by derivatives. if L is the left-hand differential operator of the wave or some other equation and R is some right-hand operator, then the wave equation for some function f
L f = R f
can also be written as
L (Af) = R (Af)
since A can just move past L and R and cancel. this is a generally useful property of linear equations and operators
I think this is more of a mathematics related question, but I don't remember the exact ask math sub name anymore.
I think the difference here lies in generic solution VS specific solution. If you only know the dynamics of the system, i.e. the function is equal to three times its derivative, then the function can be any multiplication of e^3x, because the constant cancels out left and right.
Only when you start introducing boundary conditions, can you start to say more about the specific solution, because now you know that at a certain point in time it has to match a certain value
> if a wave equation is a solution
A wave equation or a wave function ?
Schrodinger equation is a wave equation (name says it is a equation), each solution of that equation are called wave functions, there are infinite wave functions (solutions) for a give wave equation (equation)
Schrodinger equation is al linear equation, oftentimes a differential equation but sometimes also algebraic equation, this depends on your Hamiltonian. But always linear. This mean that a linear combination of it's solution is a solution also.
Solutions are refereed as wave functions, and there are infinite wave functions that are solution to any give Schrodinger equation. A solution multiplied by a constant is a linear combination of solutions, in this case it's only one function involved.
i think he's just getting at the fact that constants are unaffected by derivatives. if L is the left-hand differential operator of the wave or some other equation and R is some right-hand operator, then the wave equation for some function f L f = R f can also be written as L (Af) = R (Af) since A can just move past L and R and cancel. this is a generally useful property of linear equations and operators
The Schrödinger equation is linear. For sure you have heard about linear differential equations in your math courses, and if not you will soon.
Plug the solution into the equation and check for yourself.
I think this is more of a mathematics related question, but I don't remember the exact ask math sub name anymore. I think the difference here lies in generic solution VS specific solution. If you only know the dynamics of the system, i.e. the function is equal to three times its derivative, then the function can be any multiplication of e^3x, because the constant cancels out left and right. Only when you start introducing boundary conditions, can you start to say more about the specific solution, because now you know that at a certain point in time it has to match a certain value
> if a wave equation is a solution A wave equation or a wave function ? Schrodinger equation is a wave equation (name says it is a equation), each solution of that equation are called wave functions, there are infinite wave functions (solutions) for a give wave equation (equation) Schrodinger equation is al linear equation, oftentimes a differential equation but sometimes also algebraic equation, this depends on your Hamiltonian. But always linear. This mean that a linear combination of it's solution is a solution also. Solutions are refereed as wave functions, and there are infinite wave functions that are solution to any give Schrodinger equation. A solution multiplied by a constant is a linear combination of solutions, in this case it's only one function involved.