# The Simple Harmonic Oscillator

*Michael Fowler, University of Virginia*

### Einstein’s Solution of the Specific Heat Puzzle

The
simple harmonic oscillator, a nonrelativistic particle in a potential ½*Cx*^{2}, is a system with wide application in both classical and quantum physics. The simplest model
is a mass sliding backwards and forwards on a frictionless surface, attached to
a fixed wall by a spring, the rest position defined by the natural length of
the spring.

Many of the mechanical properties of a crystalline solid can
be understood by visualizing it as a regular array of atoms, a cubic array in
the simplest instance, with nearest neighbors connected by springs (the valence
bonds) so that an atom in a cubic crystal has six such springs attached,
parallel to the *x*, *y*, and *z* axes. Provided the
oscillations of the atoms are not too large, the springs behave well, and the
atom sees itself in a potential _{}.

Now, as the solid is heated up, it should be a reasonable
first approximation to take all the atoms to be jiggling about independently,
and classical physics, the “Equipartition of Energy”, would then assure us that at temperature *T *each atom would have on average energy
3*kT*, *k* being Boltzmann’s constant. The specific heat per atom would then be just 3*k*.

But this is *not *what
is observed! The specific heats of all
solids drop *dramatically* at low
temperatures. What’s going on here? It took Einstein to figure it out. Recall in the earlier lecture on Black Body
Radiation that at low temperatures the blue modes were frozen out because
energy could only be absorbed or emitted in quanta, photons, and the energy per
quantum was directly proportional to the frequency, so only relatively low
energy oscillators gained energy at low temperatures.

Einstein realized that exactly the same considerations must
apply to mechanical oscillators, such as atoms in a solid. He assumed each atom to be an independent
simple harmonic oscillator, and, just as in the case of black body radiation,
the oscillators can only absorb energies in quanta. Consequently, at low enough
temperatures there is rarely sufficient energy in the ambient thermal
excitations to excite the oscillators, and they freeze out, just like blue
oscillators in low temperature black body radiation. Einstein’s picture was later somewhat
refined—the basic set of oscillators was taken to be standing sound wave
oscillations in the solid rather than individual atoms (even more like black
body radiation in a cavity) but the main conclusion was not affected. In the more modern picture of sound waves in
a solid, the “elementary” sound wave, analogous to the photon, is called the *phonon*, and has energy *hf*, where *h* is again Planck’s constant, and *f* is the sound frequency.

Oscillations of molecules can usually be analyzed fairly accurately as simple harmonic oscillations, in particular the diatomic molecule. Of course, this picture breaks down for sufficiently large amplitude oscillations—eventually any molecule breaks up.

### Wave Functions for Oscillators

What kind of wave function do we expect to see in a harmonic
oscillator potential? Whatever kinetic
energy we give the particle, if it gets far enough from the origin the
potential energy will win out, and the wave will decay for the particle going
further out. We know that when a
particle penetrates a barrier of height *V*_{0},
say, greater than the particle’s kinetic energy, the wave function decreases exponentially
into the barrier, like _{}, where _{}. But the simple
harmonic oscillator potential is less penetrable than a flat barrier, because
its height increases as *x*^{2} as the particle penetrates, so we can see from the expression for a above that for large *x* a itself increases linearly in *x*. Of course, this is something of a handwaving
argument, the solution of a differential equation for a varying potential is
not just a smooth sequence of solutions for constant potentials, but it does
suggest that the right wavefunction for the oscillator potential might decay as _{}. We write it as _{}, so that the probability distribution is proportional to _{}, and *a*, which has
the dimensions of length, is a natural measure of the spread of the wave
function.

The Schrödinger equation for the simple harmonic oscillator is

_{}

If _{}, it is straightforward to verify that

_{}

Substituting this value in Schrödinger’s equation we find

_{}_{}

This equation can only be true for all* x* if the *x*^{2} terms are separately identically zero, that is,

_{}.

This fixes the wave function. Requiring the remaining terms to balance fixes the energy:

_{},

where w_{0} is the classical oscillator frequency—given
the particle mass *m* and the spring
constant *C*, the classical equation of
motion of the oscillator is

_{}.

Taking a solution of the form

_{},

gives _{}.

An important point here is that the energy is nonzero, just as it was for the square well. The central part of the wave function must have some curvature to join together the decreasing wave function on the left to that on the right. This “zero point energy” is sufficient in one case to melt the lattice—helium is liquid even down to absolute zero temperature (checked down to microkelvins!) because of this wave function spread.

### Using the Spreadsheet

The spreadsheet can be used to find the energies of the
eigenstates of the simple harmonic oscillator in a very similar way to those
for the square well. One technical
difference is that since the exponential increasing function diverges more
violently, it is almost impossible to avoid it becoming dominant at large *x*. However, provided the wave function is small over some range in *x*, in practice wave functions and
energies are given quite accurately. One
point worth noting is that just as for the square well, the quantum number for
the states is just the number of nodes, or zeros. The argument we gave for the square well
about how the extra nodes come into the wave function as the energy is
increased also works here.

For readers who have not at this point constructed the spreadsheet, which is a very educational exercise you should do at some point, you can download and play with one for the simple harmonic oscillator here: DOWNLOAD SPREADSHEET .

### Time Dependent States of the Simple Harmonic Oscillator

Working with the time independent Schrödinger equation, as
we have in the above, implies that we are restricting ourselves to solutions of
the full Schrödinger equation which have a particularly simple time dependence,
an overall phase factor _{}, and are states of definite energy *E*. However, the full time
dependent Schrödinger equation is a linear equation, so if *ψ*_{1}(*x*,*t*) and *ψ*_{2}(*x*,*t*) are solutions, so is any linear
combination *Aψ*_{1}+*Bψ*_{2}. Assuming *ψ*_{1} and *ψ*_{2} are definite
energy solutions for different energies *E*_{1} and *E*_{2}, the combination
will not correspond to a definite energy—a measurement of the energy will give
either *E*_{1} or *E*_{2}, with appropriate
probabilities. In the jargon, the
combination is not an “eigenstate” of the energy—but it is still a perfectly
good, physically realizable wave function.

It is instructive to examine a combination state of this form a little more closely. We know that for the ground state wave function,

_{}

and for the first excited state,

_{}.

Suppose we simply add terms of this type together (neglecting the overall normalization constant for now), for example

_{}.

Looking at this wave function for *t* = 0, we notice that the two terms have the same sign for *x* > 0, and opposite signs for *x* < 0. Therefore, sketching the probability
distribution for the particle’s position, it is heavily skewed to the right
(positive* x*). However, the two terms have different
time-dependent phases, differing by a factor _{}, so after time _{} has elapsed, a factor
of -1 has evolved between the terms. If
we *now* look at the probability
distribution |*ψ*|^{2}, it
will be skewed to the left. In other
words, if the state is not of definite energy, the probability distribution can
vary in time. Of course, the *total *probability of finding the
particle *somewhere* stays the same.
Note that the probability distribution swings back and forth with the period of
the oscillator. This discussion also
implies that an ordinary pendulum, which clearly swings back and forth, cannot
be in a state of definite energy!

### The Three Dimensional Simple Harmonic Oscillator

It is very simple to go from the one dimensional to the
three dimensional simple harmonic oscillator, because the potential _{} is a sum of separate *x*, *y*, *z* potentials, and consequently any
product _{}of three solutions of the one- dimensional harmonic
oscillator time independent Schrödinger equation will be a solution of the
three-dimensional harmonic oscillator, with energy the sum of the three
one-dimensional energies. So the states
are labeled with three quantum numbers, one for each direction, each can be 0,
1, 2, … If we call these three quantum
numbers *n*_{x}, *n*_{y}, *n*_{z} then from what we already know about the one
dimensional case, the energy of the three dimensional state must be _{}. For example, the
lowest energy state of the three dimensional harmonic oscillator, the zero
point energy, is _{}. Obviously, the
higher energy states are very degenerate—many sets of quantum numbers
correspond to the same state—because the energy only depends on the *sum* of the three integer quantum
numbers. Note that this degeneracy
arises from the *symmetry *of the
potential, the spring constant *k* is
the same in all three directions. If the
potential were of the form _{} for general *k*’s, there would be no degeneracy. (Such
potentials approximately describe oscillations of an atom in an anisotropic
crystal.)

Another approach to the three dimensional symmetric ½*kr*^{2} simple harmonic
oscillator is to try a separable wave function in spherical polar coordinates, _{}. This approach is
covered in detail in later courses in quantum mechanics, and is the standard
method for treating the hydrogen atom (where the potential cannot be written as
a sum of *x*, *y*, and *z *potentials). The angular functions describe the angular
momentum of the particle. Some insight can
be gained by considering the two dimensional case. Consider a pendulum swinging in the *x* direction (*z* is vertical). Now give it
a kick so it also has swing in the *y *direction. In general, it will follow an elliptical path
in the *x*, *y* plane. The right kick will
make it a circle. For the circular
orbit, the old fashioned Bohr quantization of angular momentum can be used to
find the energy levels.