Harmonic Oscillator in an Electric Field

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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}+\tfrac{1}{2}m\omega^2r^2-eE_{0}x.}

We may seprate the Hamiltonian into three terms, 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},\!} 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}+\tfrac{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}+\tfrac{1}{2}m\omega^2y^2,}

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

Note that each of these terms depends on only one coordinate, and that, in fact, 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\!} 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 H_z\!} are each the Hamiltonian of a one-dimensional harmonic oscillator. In fact, if we "complete the square" in 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,\!} we will find that it is also a one-dimensional harmonic oscillator, but with a shifted center. Let us, in fact, do this:

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}+\tfrac{1}{2}m\omega^2\left (x^2-\frac{2eE_{0}}{m\omega^2}x\right )=\frac{p_{x}^{2}}{2m}+\tfrac{1}{2}m\omega^2\left (x-\frac{eE_{0}}{m\omega^2}\right )^2-\frac{e^2E_0^2}{2m\omega^2}}

We may now easily write down the solution. If we take 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)=X(x)Y(y)Z(z),\!} then

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 X(x)=\frac{1}{2^{n_1}n_1!}\left (\frac{m\omega}{\pi\hbar}\right )^{1/4}\exp\left [-\frac{m\omega}{2\hbar}\left (x-\frac{eE_0}{m\omega^2}\right )^2\right ]H_{n_1}\left [\sqrt{\frac{m\omega}{\hbar}}\left (x-\frac{eE_0}{m\omega^2}\right )\right ],}

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 Y(y)=\frac{1}{2^{n_2}n_2!}\left (\frac{m\omega}{\pi\hbar}\right )^{1/4}e^{-m\omega y^2/2\hbar}H_{n_2}\left (\sqrt{\frac{m\omega}{\hbar}}y\right ),}

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 Z(z)=\frac{1}{2^{n_3}n_3!}\left (\frac{m\omega}{\pi\hbar}\right )^{1/4}e^{-m\omega z^2/2\hbar}H_{n_3}\left (\sqrt{\frac{m\omega}{\hbar}}z\right ).}

The energy may simply be written as 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_x+E_y+E_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 E_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 E_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 E_z\!} are the contributions to the energy from each of the harmonic oscillators. These 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_x=\left (n_1+\tfrac{1}{2}\right )\hbar\omega-\frac{e^2E_0^2}{2m\omega^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 E_y=\left (n_2+\tfrac{1}{2}\right )\hbar\omega,}

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 E_z=\left (n_3+\tfrac{1}{2}\right)\hbar\omega.}

The total energy is thus

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=\left (n_{1}+n_{2}+n_{3}+\tfrac{3}{2}\right )\hbar\omega-\frac{e^2E_{0}^{2}}{2m\omega^{2}}.}

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