Coherent States of the Simple Harmonic Oscillator
Michael Fowler, 10/14/07
What is the Wave Function of a Swinging Pendulum?
Consider a macroscopic simple harmonic oscillator, and to keep things simple assume there are
no interactions with the rest of the universe.
We know how to describe the motion using
classical mechanics: for a given initial position and momentum, classical mechanics correctly
predicts the future path, as confirmed by experiments with real (admittedly not perfect) systems.
But from the Hamiltonian we could also write down Schrödinger’s equation, and from that
predict the future behavior of the system.
Since we already know the answer from classical
mechanics and experiment, quantum mechanics must give us the same result in the limiting case
of a large system.
It is a worthwhile exercise to see just how this happens.
Evidently, we cannot simply follow the
classical method of specifying the initial position and momentum—the uncertainty principle
won’t allow it.
What we can do, though, is to take an initial state in which the position and
momentum are specified
as precisely as possible
. Such a state is called a
minimum uncertainty
state (the details can be found in my earlier lecture on the Generalized Uncertainty Principle).
In fact, the
ground state
of a simple harmonic oscillator
is
a minimum uncertainty state. This is
not too surprising—it’s just a localized wave packet centered at the origin.
The system is as
close to rest as possible, having only zeropoint motion.
What
is
surprising is that there are
excited states of the pendulum in which this ground state wave packet swings backwards and
forwards indefinitely, a quantum realization of the classical system, and the wave packet is
always one of minimum uncertainty.
Recall that this
doesn’t
happen for a
free
particle on a
line—in that case, an initial minimal uncertainty wave packet spreads out because the different
momentum components move at different speeds. But for the oscillator, the potential somehow
keeps the wave packet together, a minimum uncertainty wave packet at all times. These
remarkable quasiclassical states are called
coherent states
, and were discovered by Schrodinger
himself.
They are important in many quasiclassical contexts, including laser radiation.
Our task here is to construct and analyze these coherent states and to find how they relate to the
usual energy eigenstates of the oscillator.
Classical Mechanics of the Simple Harmonic Oscillator
To define the notation, let us briefly recap the dynamics of the
classical
oscillator: the constant
energy is
2
2
1
22
p
Ek
m
=+
x
or
()
2
2
2,
/
pm
x
m
E
k
ωω
+=
=
m
.
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The classical motion is most simply described in
phase space
, a twodimensional plot in the
variables
(
.
In this space, the point
)
,
mxp
ω
( )
,
corresponding to the position and
momentum of the oscillator at an instant of time moves as time progresses at constant angular
speed
in a clockwise direction around the circle of radius
2
mE
centered at the origin.
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 Fall '07
 MichaelFowler
 Δx, wave packet, ground state, Simple Harmonic Oscillator, minimum uncertainty

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