# Average value for the squared momentum in the harmonic oscillator

1. Aug 30, 2011

### davijcanton

Hi,

I'm trying to resolve a problem (17-2) of Pauling's book (Introduction to Quantum Mechanics ), but I'm not achieving this integration. So, I ask for your help. The problem says:

Calculate $\overline{p_{z}²}$ (where p$_{z}$ = momentum in z direction and $\overline{x}$ = average value of x for example) for a harmonic oscillator (in cylindrical coordinates) in a state ( n',m,n$_{z}$ ) represented by:

$\Psi_{n',m,n_{z}}$($\rho$,$\varphi$,z) = N e$^{im\varphi}$ e$^{-\alpha\rho²/2 }$ F$_{|m|,n'}$($\sqrt{\alpha}$ $\rho$) e$^{-\alpha_{z}z²/2}$ H$_{nz}$ ($\sqrt{\alpha_{z}}$ z)​

where:
N = const

$\alpha$ = cost = 4$\pi$²*mass/h² * (freq$_{0}$)

$\alpha$z = cost = 4$\pi$²*mass/h * (freq$_{z}$)

H$_{nz}$ = zn Hermite polynomial

F(k) = finite polynomial in k$^{|m|}$ I think this polynomial don't have larger influence in the result.

See you soon,

Davis​

2. Aug 30, 2011

### vela

Staff Emeritus
We're not here to do the problem for you. We're here to help you solve the problem, so show us how far you've gotten so far.

3. Aug 31, 2011

### davijcanton

Well. I've gotten one sum of three integrals. Say:
1º $\int$$^{\infty}_{-\infty}$ A e$^{\alpha_{z}z²}$ H$_{n_{z}}$($\sqrt{\alpha_{z}}$ z) H$_{n_{z}}$($\sqrt{\alpha_{z}}$ z)($\alpha_{z}$-$\alpha_{z}$²z²) dz

2º $\int$$^{\infty}_{-\infty}$ -4n$_{z}$$\alpha_{z}$A e$^{\alpha_{z}z²}$ z H$_{n_{z}}$($\sqrt{\alpha_{z}}$ z) H$_{n-1}$($\sqrt{\alpha_{z}}$ z) dz

3º $\int$$^{\infty}_{-\infty}$ 2n$_{z}$(n$_{z}$-1) e$^{\alpha_{z}z²}$ H$_{n_{z}}$($\sqrt{\alpha_{z}}$ z) H$_{n_{z}-2}$($\sqrt{\alpha_{z}}$ z) dz

In fact, I would like to know if I got this rightly and how can I resolve that.

4. Aug 31, 2011

### vela

Staff Emeritus
Use the recurrence relations the Hermite polynomials satisfy to eliminate the factors of z, and then use the orthogonality of the polynomials to evaluate the integrals.

5. Aug 31, 2011

### davijcanton

In this case, is it normal the average value for x and for p vanish?

6. Aug 31, 2011

### vela

Staff Emeritus
What do you think? Is that what you'd expect physically?