# Harmonic osilator energy using derivatives

1. Dec 12, 2013

### dawozel

1. The problem statement, all variables and given/known data

Show that the energy of a simple harmonic oscillator in the n = 2 state is 5Planck constantω/2 by substituting the wave function ψ2 = A(2αx2- 1)e-αx2/2 directly into the Schroedinger equation, as
broken down in the following steps.

First, calculate dψ2/dx, using A for A, x for x, and a for α.

Second, calculate d2ψ2/dx2.

3. The attempt at a solution
so I got the first derivative correct, it was

A((4*a*x)*exp((-a*x^2)/2) +(2*a*x^2 -1)*(-a*x*exp((-a*x^2)/2)))

but i can seem to calculate the second derivative correctly I'm getting
$A((4a)(exp(( - ax^2 ) / 2)) + (4ax) * ( - ax * exp(( - ax^2) /2)) + (4ax) * ( - a xexp(( - ax^2 ) / 2)) + (2ax - 1) (a^2x^2 *exp(( - ax^2 ) / 2)))$

but this incorrect, am I missing something?

Last edited: Dec 12, 2013
2. Dec 12, 2013

### Simon Bridge

$$\psi_2(x)=A(2\alpha x^2-1)e^{-\frac{1}{2}\alpha x^2}$$
$$\frac{d}{dx}\psi_2(x)=A\left[4\alpha x e^{-\frac{1}{2}\alpha x^2} - (2\alpha x^2 - 1)\alpha x e^{-\frac{1}{2}\alpha x^2}\right]$$... is pretty messy so it will be easier to make mistakes: simplify this expression first. Then try the second derivative.

You can bring an $\alpha x \exp(-\frac{1}{2}\alpha x^2)$ outside the brackets ... then deal with the terms inside the brackets.

After that I suspect it is something you can do.

note: use the "quote" button below this post to see how I got the equations to typeset like that ;)