Average value for the squared momentum in the harmonic oscillator

[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.



Thenks in advance.
See you soon,

Davis​

 

[vela]

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.

 

[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.

 

[vela]

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.

 

[davijcanton]

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

 

[vela]

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