Perturbation theory in 3D potential

[JayKo]

Homework Statement

Consider a quantum particle of mass m in a 3-D harnonic potential with frequency \omega and it experiences a perturbation H_{1}=az^{2}

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

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

Homework Equations

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

The Attempt at a Solution

Subsititute in the perturb into the energy equation.

E^{(1)}=<\Psi|az^{2}|\Psi>

1st excited state is |\Psi_{112}>, |\Psi_{121}>, |\Psi_{211}>

then find the value of <\Psi_{112}|az^{2}||\Psi_{112}>, <\Psi_{121}|az^{2}||\Psi_{121}>,
<\Psi_{211}|az^{2}||\Psi_{211}>, susb z^{2}=r^{2}-x^{2}-y^{2}
and x^{2}=\frac{\hbar}{2m\omega}[a^{2}_{+}+a_{+}a_{-}+a_{-}a_{+}+a^{2}_{-}]
but i don't know what is the representation for y^{2} and r^{2}

 

[JayKo]

can anyone tell me if i am heading the right direction? thanks