Solving the Schroedinger Equation for An Anisotropic Oscillator Potential

[Wishe Deom]

Homework Statement

Consider a particle of mass m moving in a 3D-anisotropic oscillator potential:
$$V(\vec{r}) = \frac{1}{2}m(\omega^{2}_{x}x^{2}+\omega^{2}_{y}y^{2}+\omega^{2}_{z}z^{2})$$. (a) Frind the stationary states for this potential and their respective energies.

Homework Equations

Time-Independent Schroedinger Equation in 3 dimensions is $$\frac{\bar{h}^{2}}{2m}\nabla^{2}\psi+V\psi=E\psi$$

The Attempt at a Solution

I first tried to find solutions to the TISE in the form of $$\psi=X(x)Y(y)Z(z)$$, taking all the partial derivatives, dividing through by XYZ, and arranging one side to be a function of x, and the other to be a function of y and z, equaling a constant of separation A.

However, when solving for X(x), I have an equation of the form $$\frac{\partial^{2}X}{\partial x^2} = (C + x^2}X$$. I have no idea how to solve this for X. Am I apporaching this problem in the correct way?

 

[diazona]

You'd know how to solve that equation if C were equal to zero, right? Try starting with that solution, and then see if you can add something to it to make it satisfy the equation you have.

 

[Wishe Deom]

You'd know how to solve that equation if C were equal to zero, right? Try starting with that solution, and then see if you can add something to it to make it satisfy the equation you have.

I didn't know the solution to X'' = x^2 X, but Wolfram Alpha tells me it involves some function D, which I have never before seen. That's why I was having doubts as to whether my reasoning up to this point had been sound.

 

[fzero]

Check all of your signs and then review the solution of the 1d harmonic oscillator. The correct equations give 3 copies of the Schrodinger equation for the 1d HO.

 

[diazona]

Oh wait, I think I misread the equation (and was thinking about entirely the wrong problem!). Yes, you're right, solving that equation isn't quite trivial.

How much do you know about the one-dimensional quantum harmonic oscillator?

 

[Wishe Deom]

I am very familiar with the 1d harmonic oscillator, and I would know how to solve each of the three equations if I got them, but I'm not sure how to get there.

If I look for solutions of the form $$\psi = X(x) + Y(y) +Z(z)$$, then the term in the SE $$V \psi$$ would be some nine-term monstrosity, wouldn't it?

 

[fzero]

You were correct to look for solutions in terms of a product. Rewrite the equation for $$X(x)$$ (it would help to leave the physical quantities in) and compare it to the equation for a 1d HO.

 

[diazona]

I am very familiar with the 1d harmonic oscillator, and I would know how to solve each of the three equations if I got them, but I'm not sure how to get there.

What's the difference between the equation you got for the x direction and the equation for the 1D harmonic oscillator?

 

[dingo_d]

Try assuming the solution of the form:

$$X(x)=e^{\frac{x^2}{2}}v(x)$$ and put it into the original DE equation (for X(x)). You should get the DE for v(x) which you can solve by power series method. And there should be some constrains on the recursion...