Help with solving a differential equation

[A Tripolation]

Ok. I'm having a very difficult time with this problem. I can't even do this first part. Any help would be greatly appreciated. Like what steps I should take, what method to use, whether or not to become a lit. major, ect., ect.

Here it is. Part 1.

 

We seek to solve the time-independent Schrodinger equation:

 

[math] \frac{-\hbar^2}{2m} \frac{d^2\psi}{dx^2} + \frac{1}{2}m\omega^2 x^2 \psi = E\psi [/math]

 

First, introduce the dimensionless variable:

 

[math] \xi = \sqrt{ \frac{m \omega}{\hbar}}x, [/math]

 

and show that the Schrodinger equation becomes:

 

[math] \frac{d^2 \psi}{d \xi^2} = (\xi^2 - K) \psi, [/math]

 

where K is the energy in units of [math] \frac{1}{2} \hbar \omega,[/math] say [math] K = \frac{2 \xi}{\hbar \omega} [/math]

 

 

Sooooo...what about it?

Edited September 30, 2010 by A Tripolation

[the tree]

Looks to me like the second form is seperable. Of course, I just misread what you were asking for. Sorry about that.

Edited October 1, 2010 by the tree

[A Tripolation]

Looks to me like the second form is seperable. Of course, I just misread what you were asking for. Sorry about that.

 

I'm just trying to figure out how to do it. I don't see how to "introduce the dimensionless variable".

[timo]

I'd not care about the exact wording, not "introduce xi", and instead show the equivalence the other way round, i.e. starting with the expression involving xi and working back from there. It's about 5 lines of handwriting.

[A Tripolation]

I'd not care about the exact wording, not "introduce xi", and instead show the equivalence the other way round, i.e. starting with the expression involving xi and working back from there. It's about 5 lines of handwriting.

 

Ohhhhhhhhhh. I see now. Thanks to you timo. And you too tree.

Edited October 2, 2010 by A Tripolation

[A Tripolation]

So I've reached Part Three in this problem. By some crazy luck, I've managed to reduce the equations to:

 

[math] \psi (x) = C_1 e^{\xi^2/2} + C_2 e^{-\xi^2/2} [/math]

 

The term, [math] C_1 e^{\xi^2/2} [/math] grows impossibly large as [math] \xi\to\infty [/math] , so it can be thrown out as a physically implausible term, correct?

 

If so, I get, [math] \psi(\xi) \approx C_2 e^{-\xi^2/2} [/math], which is [math] \psi(\xi) = h(\xi)e^{-\xi^2/2} [/math], based on other information given in the problem.

 

But, h is an unknown function. I'm supposed to find it somehow. I'm thinking power series, maybe, but do not know if that's correct or how to get started on that.

 

Ideas?