# Simple Harmonic Oscillator - Normalization Constant

1. Oct 8, 2009

### atclaeys

1. The problem statement, all variables and given/known data
Determine the normalization constants for the harmonic oscillator wavefunctions with v=0, and v=1 by evaluating their normalization integrals and show that they correspond to N=$$\frac{1}{\pi^{.5} * 2^v * v!}$$

2. Relevant equations

3. The attempt at a solution

$$\int \psi^{2}d\tau$$=1
$$\psi$$1=N*2y*exp(-y$$^{2}$$/2)
$$\psi^{2}$$=N2*4y$$^{2}$$exp(-y$$^{2}$$)
$$\int \psi^{2}d\tau$$ = $$\int N^{2}4y^{2}e^{-y^{2}}dy$$=1

By a table of integrals, integral from 0->inf of y$$^{2n}$$exp(-ay$$^{2}$$) => ($$\frac{(2n!)\pi^{.5}}{2^{2n+1}n!a^{n+1/2}}$$)

So what I end up with is N1=$$\sqrt{\frac{1}{\pi^{.5}}}$$

The given equation of N I evaluate to be $$\sqrt{\frac{1}{2\pi^{.5}}}$$

I've looked through my integration several times, but am not sure where I could be coming up with an error (and checked the formula given by table with various other tables), any help is appreciated.

EDIT: Noticed that in cartesian, all space is -inf to inf, not 0 to inf as in spherical polar....

Last edited: Oct 8, 2009