Simple Harmonic Oscillator - Normalization Constant

[atclaeys]

Homework Statement

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=$$\frac{1}{\pi^{.5} * 2^v * v!}$$

Homework Equations

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...

 

[NateD]

so my integral should be from -inf to inf, which is correct for the given equation of N, so that is the issue.