Harmonic Oscillator in an Electric Field

consider a particle with charge e moving under three dimensional isotropic harmonic potential l

$V(r)={\frac {1}{2}}m{\omega }^{2}{r}^{2}$in an electric field $E=E_{0}(x)$ Find the eigen states and eigen values of the patricle

the Hamiltonian of the system is:

$H={\frac {P^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}r^{2}-eE_{0}x$

we seprate the Hamiltonian ($H=H_{x}+H_{y}+H_{z}f$) where

$H_{x}={\frac {p_{x}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}x^{2}-eE_{0}x$

$H_{y}={\frac {p_{y}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}y^{2}$

$H_{z}={\frac {p_{z}^{2}}{2m}}+{\frac {1}{2}}m\omega ^{2}z^{2}$

Notice that $H_{x},H_{z}$are identical to the Hamiltonian of the one dimensional harmonic oscillator, so we can write the wave function

$\psi (x,y,z)=\psi _{1}(x)\psi _{2}(y)\psi _{3}(z)$, whereFailed to parse (unknown function "\math"): {\displaystyle \psi _{2}(y)<\math>, and }
\psi _{3}(z) </math>are the wave functions of the one dimensional harmonic oscillator: