# Calculating a Quantum Nomalization Constant

1. Oct 16, 2008

### Seda

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

If I have a wave function that = Ce((-mwx2)/(2*hbar))

(Its the wave function of the ground state of a simple harmonic oscilator)

How do I calculate C?

2. Relevant equations

Quantum Normalization condition I think is all i need.

3. The attempt at a solution

C2 is pulled out of the integral because its a constant

Leaving me with the integral from - infinity to infinity of e-mwx2/hbar

How do I integrate that? Is there an easier way to solve for C?

Everythings 1D

2. Oct 16, 2008

### Niles

3. Oct 16, 2008

### Seda

Ok, now I know that

C= sqrt[1/(sqrt((pi*hbar)/(m*w)))]

However, I dont know w.

Odd....

4. Oct 16, 2008

### Seda

Ok, now I need to solve for <x2>

Which means I obviously end up with:

C2 times the integral from - infinity to inifinty of x2*e-ax2

where a = mw/hbar

I can seem to find a solution to this integral in my handbook. How do you intgrate that?

5. Oct 17, 2008

### Niles

Last edited: Oct 17, 2008
6. Oct 17, 2008

### DeShark

7. Oct 17, 2008

### Niles

\begin{align*} \int_{ - \infty }^\infty {x^2 \exp \left( { - \lambda x^2 } \right)} dx = - \int_{ - \infty }^\infty {\frac{\partial }{{\partial \lambda }}\exp \left( { - \lambda x^2 } \right)} dx = - \frac{\partial}{{\partial\lambda }}\int_{ - \infty }^\infty {\exp \left( { - \lambda x^2 } \right)} dx = - \frac{d}{{d\lambda }}\sqrt {\frac{\pi }{\lambda }} \end{align*}

- this is what the link contains. It is called differentiation under integration.

Of course partial integration can be used too, but the above method simplifies things greatly.