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Average value for the squared momentum in the harmonic oscillator

  1. Aug 30, 2011 #1
    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 [itex]\overline{p_{z}²}[/itex] (where p[itex]_{z}[/itex] = momentum in z direction and [itex]\overline{x}[/itex] = average value of x for example) for a harmonic oscillator (in cylindrical coordinates) in a state ( n',m,n[itex]_{z}[/itex] ) represented by:

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

    where:
    N = const

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

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

    H[itex]_{nz}[/itex] = zn Hermite polynomial

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



    Thenks in advance.
    See you soon,

    Davis​
     
  2. jcsd
  3. Aug 30, 2011 #2

    vela

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    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.
     
  4. Aug 31, 2011 #3
    Well. I've gotten one sum of three integrals. Say:
    1º [itex]\int[/itex][itex]^{\infty}_{-\infty}[/itex] A e[itex]^{\alpha_{z}z²}[/itex] H[itex]_{n_{z}}[/itex]([itex]\sqrt{\alpha_{z}}[/itex] z) H[itex]_{n_{z}}[/itex]([itex]\sqrt{\alpha_{z}}[/itex] z)([itex]\alpha_{z}[/itex]-[itex]\alpha_{z}[/itex]²z²) dz

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

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


    In fact, I would like to know if I got this rightly and how can I resolve that.
     
  5. Aug 31, 2011 #4

    vela

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    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.
     
  6. Aug 31, 2011 #5
    In this case, is it normal the average value for x and for p vanish?
     
  7. Aug 31, 2011 #6

    vela

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