Quantum mechanics hermite polynomials

[jc09]

Homework Statement

Show that the one-dimensional Schr¨odinger equation
ˆ
(p^2/2m) ψ+ 1/2(mw^2)(x)ψ = En ψ
can be transformed into
(d^2/d ξ ^2)ψ+ (λn- ξ ^2) ψ= 0 where λn = 2n + 1.
using hermite polynomials

Homework Equations

know that dHn(X)/dX= 2nHn(x)

The Attempt at a Solution

I don't know how to start this question off at all

 

[quZz]

The question is a bit unclear to me. If I understand it correctly you should do the following:
1. get the first equation to the form
$$d^2\psi/dx^2 + 2m/\hbar^2(E-m\omega^2 x^2/2) \psi = 0$$
2. substitute $$\xi = x \sqrt{m\omega/\hbar}$$
3. you should get
$$d^2\psi/d\xi^2 + (2E/\hbar \omega - \xi^2) \psi = 0$$
4. now substitute $$\psi = \varphi(\xi) \exp(-\xi^2/2)$$ to get
5. $$d^2\varphi/d\xi^2 - 2\xi d\varphi/d\xi + (2E/\hbar\omega-1)\varphi = 0$$
This equation has solutions that diverge at infinity not faster than a polynomial - Hermite polynomials - only when $$2E/\hbar\omega-1 = 2n$$, where $$n = 0,1,2,...$$

Hope that helps!

 

[jc09]

Hey that's perfest thanks was stuck on how to get ξ into the equation.