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