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The Harmonic Oscillator
Now consider a particle subject to a restoring force , as might
arise for a massspring system obeying Hooke's Law. The potential is
then
If we choose the energy scale such that then
. This potential is also appropriate for describing the
interaction of two masses connected by an ideal spring. In this case,
we let be the distance between the masses, and for the mass we
substitute the reduced mass . Thus the harmonic oscillator is
the simplest model for the vibrational motion of the atoms in a
diatomic molecule, if we consider the two atoms as point masses and
the bond between them as a spring. The onedimensional Schrödinger
equation becomes

(119) 
After some effort, the eigenfunctions are

(120) 
where is the Hermite polynomial of degree , and and
are defined by

(121) 
The eigenvalues are

(122) 
with
.
Next: The Rigid Rotor
Up: Some Analytically Soluble Problems
Previous: The Particle in a
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David Sherrill
20060815