Harmonic Oscillator in an Electric Field

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Source: "Theory and problems of quantum mechanics", Schaum, chapter 5

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 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 \xi =\frac{x}{\lambda }-\frac{eE_{0}}{\sqrt{\hbar m \omega }}}

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{\partial^2 \psi _{1}}{\partial \xi ^2}+(\frac{2E_{1}}{\hbar \omega })\frac{(eE_{0})^{2})}{\sqrt{\hbar m\omega ^{3}}})\psi _{1}-\xi ^{2}\psi _{1}=0}

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}(\xi )=\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}] (E_{1})_{n_{1}}=(n_{1}+\frac{1}{2})\hbar \omega -\frac{(eE_{0})^{2}}{2m\omega ^{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}}=(n_{1}+\frac{1}{2})\hbar \omega -\frac{(eE_{0})^{2}}{2m\omega ^{2}}}

In conclusion,the wave functions 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 \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 E_{n_{1},n_{2},n_{3}}=E_{n_{1}}+E_{n_{2}}+E_{n_{3}}=(n_{1}+n_{2}+n_{3}+\frac{3}{2})\hbar \omega -\frac{(eE_{0})^{2}}{2m\omega ^{2}}}

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