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\pi²*mass/h² * (freq_{0})

\alphaz = cost = 4\pi²*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?