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Now consider a particle subject to a restoring force F = kx, as might
arise for a massspring system obeying Hooke's Law. The potential is
then
V(x) 
= 

(118) 

= 


If we choose the energy scale such that V_{0} = 0 then
V(x) =
(1/2)kx^{2}. This potential is also appropriate for describing the
interaction of two masses connected by an ideal spring. In this case,
we let x be the distance between the masses, and for the mass m 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 H_{n} is the Hermite polynomial of degree n, and
and
N_{n} are defined by

(121) 
The eigenvalues are

(122) 
with
.
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Up: Some Analytically Soluble Problems
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