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
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