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:

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=\frac{P^2}{2m}+\frac{1}{2}m\omega ^2r^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}

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

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 \psi (x,y,z)=\psi _{1}(x)\psi _{2}(y)\psi _{3}(z)}
, 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 \psi _{2}(y)} , and 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 \psi _{3}(z)} are the wave functions of the one dimensional harmonic oscillator: 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 \psi (x,y,z)=\psi _{1}(x)\psi _{2}(y)\psi _{3}(z)}

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 \psi _{2}(y)=\frac{1}{\sqrt{\pi \lambda 2 ^{n_{2}}n_{2}!}}H_{n_{2}}e^{\frac{-y^{2}}{2\lambda ^{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 \psi _{3}(z)=\frac{1}{\sqrt{\pi \lambda 2 ^{n_{3}}n_{3}!}}H_{n_{3}}e^{\frac{-z^{2}}{2\lambda ^{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 \lambda =\sqrt{\frac{\hbar}{m\omega }},} The equation of theFailed 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 _{1}(x)} is

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 -\frac{\hbar^{2}}{2m}\frac{\partial^2\psi _{1}(x)}{\partial x^2}+\frac{m\omega ^{2}}{2}x^{2}\psi _{1}(x)-eE_{0}(x)\psi _{1}(x)=E_{1}\psi _{1}(x)} changing variables to

we obtain the diffrential equation for a one dimensional harmonic oscillator with the solution

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 \psi _{1}(\xi )=\frac{1}{\sqrt{\pi \lambda 2 ^{n_{1}}n_{1}!}}H_{n_{1}}e^{\frac{-\xi ^{2}}{2\lambda ^{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 \psi _{1}(x )=\frac{1}{\sqrt{\pi \lambda 2 ^{n_{1}}n_{1}!}}H_{n_{1}(x)}exp(\frac{-1}{2}(\frac{x}{\lambda }-\frac{eE_{0}{\sqrt{\hbar m\omega^{3}})^2}})}

The quantization condition in this case is 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 \frac{(2E_{1})}{\hbar\omega }+\frac{(eE_{0})^2}{\hbar m\omega ^{3}}=2n_{1}+1} so the energy eigenvalues are 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_{1}=(n_{1}+\frac{1}{2})\hbar}