Harmonic Oscillator in an Electric Field

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)=\frac{1}{2}m{\omega }^2{r}^2} in an electric field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=H_{x}+H_{y}+H_{z} f} ) where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{x}=\frac{p_{x}^{2}}{2m}+\frac{1}{2}m\omega ^2x^2-eE_{0}x}

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{z}=\frac{p_{z}^{2}}{2m}+\frac{1}{2}m\omega ^2z^2}

Notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi _{2}(y)<\math>, and }
\psi _{3}(z) </math>are the wave functions of the one dimensional harmonic oscillator: