Wave Equation in 2 Dimensions - Basic

[DougD720]

Hey Everyone,

So I've been working on some very basic QM mathematics. Basically I've worked out the wave equation for a particle in one dimension (briefly) like so:

-[tex]\frac{\hbar [SUP]2[/SUP]}{2m}[/tex][tex]\psi[/tex]"(x) + V(x)[tex]\psi[/tex](x) = E[tex]\psi[/tex](x)

V = 0 for 0 < x < L  ; (L = "Length" of the Boundary)

=> [tex]\psi[/tex](x) = A sin([tex]\frac{n \pi x}{L}[/tex])

=> A = [tex]\frac{L}{2}[/tex]

The trouble I'm having is trying to extrapolate this to two spatial dimensions (if that can be done in the fashion I'm trying).

I follow the same process except my solution to the Schrödinger equation (solution to the differential equation) is

[tex]\psi[/tex](x,y) = A sin([tex]\frac{n \pi x}{L}[/tex]) + B sin([tex]\frac{n \pi y}{L}[/tex])

A[SUP]2[/SUP]([tex]\frac{L}{2}[/tex])y + B[SUP]2[/SUP]([tex]\frac{L}{2}[/tex])x = 1

^ From Normalizing the Solution with Limits of integration for the double integral of 0 < (x,y) < L

The problem is that instead of finding the 'constants' A and B I've now got a relationship between them.

If someone could point out what I did wrong in my process (determine the wave equation differential equation, normalize and solve) and if that's alright where I go from here in writing the full wave-function of the model, I'd greatly appreciate it.

Thanks!

 

[Pengwuino]

Why do you think you can simply add solutions? You can use the method of separation of variables to find solutions of the form \psi (x,y) = X(x)Y(y) however. To be sure, plug your solution into the equation and you'll see that it isn't a solution.

 

[DougD720]

Thank you, I'll work that one out

 

[DougD720]

I'm in need of some direction... I have differential equation experience and partial integrals/derivatives but I believe this is a partial differential equation now that both the x and y variables have been introduced and I have not worked on partial diff eqs. Where do I start to tackle this one? I tried a few solutions with a guess and check and none of them have worked and I'm a bit stuck with this one. Any help would be appreciated, thank you

 

[Pengwuino]

Well, you should find a section on Separation of Variables in your QM text if you have one or a book on PDE's. You seem like you have an understanding enough for me to just say the following: Look for solutions of the form \psi (x,y) = X(x)Y(y) . Your Hamiltonian now is the 2-D Laplacian so your DE looks like \frac{{ - h^2 }}{{2m}}(\frac{{\partial ^2 }}{{\partial x^2 }} + \frac{{\partial ^2 }}{{\partial y^2 }})\psi (x,y) = E\psi (x,y). There's some math that shows the solution form we have is valid so you can check that out on your own.

Now simple arranging shows this is also \frac{{\partial ^2 \psi (x,y)}}{{\partial x^2 }} + \frac{{\partial ^2 \psi (x,y)}}{{\partial y^2 }} = \frac{{ - 2mE}}{{h^2 }}\psi (x,y). At this point, you plug in your solution \psi (x,y) = X(x)Y(y). At this point you can do some slight manipulations and what you'll get is basically 2 separated ODE's summing to 0. For PDE's, the only way this is possible is if both ODE's are equal to a constant (otherwise you couldn't have independent variations between x and y which is a requirement from your DE). Make up a set of constants, say m and -m and from there, solve the ODE's and look back on your solution form and construct your wavefunction.

 

[DougD720]

I don't have a QM text but I know of separation of variables from diff. eq. I have to take the time to read through your explanation but from skimming it it looks like I can follow your steps. Thank you!