Harmonic Oscillator in an Electric Field: Difference between revisions

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Source: "Theory and problems of quantum mechanics", Schaum, chapter 5
The Hamiltonian of the system is:
:'''consider a particle with charge e moving under three dimensional isotropic harmonic potential '''l
 
<math>V(r)=\frac{1}{2}m{\omega }^2{r}^2</math>'''in an electric field'''  <math>E=E_{0}(x)</math> '''Find the eigen states and eigen values of the patricle'''
 
the Hamiltonian of the system is:


<math>H=\frac{P^2}{2m}+\frac{1}{2}m\omega ^2r^2-eE_{0}x</math>
<math>H=\frac{P^2}{2m}+\frac{1}{2}m\omega ^2r^2-eE_{0}x</math>

Revision as of 15:38, 9 August 2013

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

, where

, and are the wave functions of the one dimensional harmonic oscillator:

The equation of the is

changing variables to

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

The quantization condition in this case is so the energy eigenvalues are

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 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}}}