6210/Spring Physics 2007/Lecture 16
Lecture 16
Relevant sections in text: 2.2, 2.3
Ehrenfests Theorem
The Heisenberg equations are appealing because they make formal contact with the
Hamilton equations of classical mechanics. In classical mechanics functions on phase space
represent the observables, and the time rate of change of an observable A is controlled by
the Poisson bracket with the Hamiltonian:
dA
= {A, H }.
dt
The formal correspondence with quantum mechanics is made via
{A, B }
1
[A, B ],
i
h
where the observables are represented by functions on phase space on the left and operators
on the right. This formal correspondence implies that expectation values will, in a suitable
approximation, follow classical trajectories, a result known as Ehrenfests theorem.
To derive this theorem in the Heisenberg picture is quite easy. Take the expectation
value of the quantum form of Newtons second law,
d2 X i (t)
V
= i (X(t)),
dt2
x
and use the time independence of the state vector to obtain (exercise)
d2
X (t) = F (t),
dt2
where F is the force. This result is Ehrenfests theorem.
Exercise: How would you derive this equation in the Schrdinger picture?
o
It is often said that Ehrenfests theorem shows that expectation values obey the classical dynamical laws. This slogan is not quite true. In particular, the expectation value
of position does not necessarily obey Newtons second law. A true version of Newtons
second law for the expectation value would read
m
d2 X i (t)
V ( X (t))
=
.
2
dt
Xi
1
Physics 6210/Spring 2007/Lecture 16
Of course, this latter equation is not what quantum mechanics gives, in general. To get
this last result we need
V (X (t))
V ( X (t))
=
,
X i
Xi
whose validity depends upon the state vector being used as well as on the form of the
potential energy operator. A simple example where this equality does hold is provided by
the harmonic oscillator potential in one dimension (to be discussed below in more detail)
for which
1
V (X ) = kX 2 ,
2
so that
V (X (t))
= kX,
X
which satises for any state vector
k X (t) =
V ( X (t))
= k X (t).
X
The harmonic oscillator example is exceptional. Generally, Ehrenfests theorem does
not imply that expectation values obey classical equations of motion. For example, consider
the potential
1
V (X ) = kX 3 .
3
We have in this case
V ( X (t))
= k X 2 (t),
X
which does not, in general, equal
V (X (t))
= k X 2 (t).
X
Indeed, these two expressions agree if and only if the dispersion in X (t) vanishes which it
never does, strictly speaking. (On the other hand, one can nd states which make the position dispersion arbitrarily small at the expense of a large dispersion in momentum.) Thus,
according to Ehrenfests theorem, the expectation values of position for this cubic potential
will only agree with the classical behavior insofar as the dispersion in position is negligible
(for all time) in the chosen state. This result illustrates a general rule: classical behavior
for particle motion arises when the statistical uncertainties in the basic observables are
suciently small.
Harmonic Oscillator: Denitions, the Hamiltonian
We now begin a survey of key properties of one of the workhorse models of quantum
mechanics: the simple harmonic oscillator (SHO). This model is useful because it is analytically quite tractable, it illustrates a wide variety of quantum mechanical features both
2
Physics 6210/Spring 2007/Lecture 16
physically and at the level of the formalism itself, and it provides a very nice rst approximation to many simple models of real physical systems in the vicinity of stable equilibrium.
It is remarkable how often the idea of a harmonic oscillator appears in physics. For example, the harmonic oscillator generic illustrates features of bound state behavior, it
is used to model the connement of quarks in nucleons, it is used to model the spectral
properties of molecules, it is the elementary building block of the free quantum eld.
One can use harmonic oscillators to understand photons!
The SHO can be viewed mathematically as a particle moving in one dimension under
the inuence of a quadratic potential (although, physically, it doesnt usually arise that
way). The position is X , the momentum is P , which we have already dened in some
detail (in the Heisenberg pictures these are the position and momentum operators at the
initial time). The dynamics are generated by the Hamiltonian (in either the Schrdinger
o
or Heisenberg picture)
P2
1
H=
+ m 2 X 2 .
2m 2
Classically, this would be the energy function for a mass on a spring dynamical system
where m is the mass and is the natural frequency of the spring according to Hookes
law. We retain that terminology here.
The position and momentum observables we have studied in some detail. For this
simple one-dimensional system, the Hamiltonian is the center of attention. Of course, we
would like to understand its spectrum and eigenvectors, since these characterize the possible energies and the states where the energy is known with statistical certainty. Moreover,
the spectral properties of H will determine the dynamical evolution of the oscillator in the
Schrdinger picture. The algebraic (commutator) properties of H , X and P will control
o
the dynamics in the Heisenberg picture. In fact, as you will see, the algebraic properties
of the operators H , X and P essentially tell us everything we want to know.
Spectrum of the SHO Hamiltonian
To analyze H we follow an elegant algebraic approach pioneered (I believe) by Dirac.
Rather than working directly with the position and momentum operators, a more convenient set of operators can be dened as follows. Set
a=
m
2
h
i
P
m
,
h
(a + a ),
2m
P=
m
2
h
a =
X+
X
i
P
m
so that (exercises)
X=
3
1
i
h
(a a ).
2m
,
Physics 6210/Spring 2007/Lecture 16
Note that a and a can be viewed as quantum versions of the complex amplitudes featuring in the general solution of the equations of motion of a classical oscillator (exercise).
We will say a little more about this later.
From the CCR for position and momentum, we have (exercise)
[a, a ] = I.
A straightforward computation reveals (exercise)
1
1
h
h
H = (aa + a a) = (a a + I ).
2
2
Evidently, the gist of the Hamiltonian is to be found in the number operator
N := a a.
This operator is self-adjoint, and we denote its eigenvectors, for now, by | ,
N | = | ,
so that the energy eigenvectors are | with
1
h
E = ( + ) .
2
Note that the eigenvalues are dimensionless (exercise).
H | = E | ,
It is not hard to check that*
[N, a ] = a .
[N, a] = a,
This implies that (exercise)
N (a | ) = ( + 1)(a | ),
and
N (a| ) = ( 1)(a| ).
Therefore a (a) takes an eigenvector of N and turns it into an eigenvector with eigenvalue
raised (lowered) by one unit. Correspondingly, the action of a (a) on an energy eigenvector
yields an eigenvector with energy raised (lowered) by one quantum, h . For this reason
and a are often called creation and annihilation operators, or ladder operators or
a
raising and lowering operators. So, if one has a single eigenvector of N (or H ), then one
will have an innity of such eigenvectors with eigenvalues all diering by integers (or by
integer multiples of ). This result follows merely from the commutation relations for
h
position and momentum (or a and a ).
To be continued. . .
* One way to check this is to use the identity
[AB, C ] = A[B, C ] + [A, C ]B,
which you should prove as an exercise.
4