Harmonic Oscillator in an Electric Field

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The Hamiltonian of the system is

We may seprate the Hamiltonian into three terms, where

and

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

We may now easily write down the solution. If we take then

and

The energy may simply be written as where and are the contributions to the energy from each of the harmonic oscillators. These are

and

The total energy is thus

Back to Analytical Method for Solving the Simple Harmonic Oscillator