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Perturbation theory in 3D potential

  1. Oct 21, 2009 #1
    1. The problem statement, all variables and given/known data
    Consider a quantum particle of mass m in a 3-D harnonic potential with frequency [itex]\omega[/itex] and it experiences a perturbation [itex]H_{1}=az^{2}[/itex]

    a. Determine the effect of [tex]H_{1}[/tex] on the 1st exicted level of the system ( at the 1st order perturbation)

    b. what happen to L[tex]^{2}[/tex] and [tex]L_{z}[/tex]? are they still conserved in presence of [tex]H_{1}[/tex]?

    2. Relevant equations

    1st order : [itex]E^{(1)}=<\Psi|H_{1}|\Psi[/itex]>

    3. The attempt at a solution

    Subsititute in the perturb into the energy equation.

    [itex]E^{(1)}=<\Psi|az^{2}|\Psi[/itex]>

    1st excited state is [itex]|\Psi_{112}[/itex]>, [itex]|\Psi_{121}[/itex]>, [itex]|\Psi_{211}[/itex]>

    then find the value of <[itex]\Psi_{112}|az^{2}||\Psi_{112}[/itex]>, <[itex]\Psi_{121}|az^{2}||\Psi_{121}[/itex]>,
    <[itex]\Psi_{211}|az^{2}||\Psi_{211}[/itex]>, susb [itex]z^{2}=r^{2}-x^{2}-y^{2}[/itex]
    and [itex]x^{2}=\frac{\hbar}{2m\omega}[a^{2}_{+}+a_{+}a_{-}+a_{-}a_{+}+a^{2}_{-}][/itex]
    but i don't know what is the representation for [itex]y^{2} and r^{2}[/itex]
     
    Last edited: Oct 21, 2009
  2. jcsd
  3. Oct 21, 2009 #2
    can anyone tell me if i am heading the right direction? thanks
     
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