Mechanics: Quantum Commutation
5 april 2010
I. Commutators: Measuring Several Properties Simultaneously
In classical mechanics, once we determine the dynamical state of a system,
we can simultaneously obtain many dierent system properties (i.e., velocity, position, momentum, acceleration, angular/linear momentum, kinetic
and potential energies, etc.). The uncertainty is governed by the resolution
and precision of the instruments at our disposal.
In quantum mechanics, the situation is dierent. Consider the following:
1. We would like to measure several properties of a particle represented by a wavefunction.
2. Properties of a q.m. system can be measured experimentally. Theoretically, the measurement process corresponds to an operator acting on
the wavefunction. The outcomes of the measurement are the eigenvalues
that correspond to the operator. The operator is taken to be acting on a
wavefunction that is either a pure eigenfunction of the operator of interest,
or an expansion in the basis of eigenfunctions.
In order to measure, for instance, 2 properties simultaneously, the wavefunction of the particle must be an eigenstate of the two operators
that corespond to the properties we would like to measure simultaneously. How do we formalize this mathematically?
Consider:
We have to operators, A and B . Each operator acting on its eigenstate
gives back the corresponding eigenvalues, A i and Bj , respectively.
A Ai = A i Ai
B Bi = B i Bi
If we have a wavefunction that is an eigenstate of both operators, then:
AAi ,Bj = Ai Ai ,Bj
1
BAi ,Bj = Bj Ai ,Bj
Thus,
B AAi ,Bj = BAi Ai ,Bj
ABAi ,Bj = ABj Ai ,Bj
So, using the fact that Ai ,Bj is an eigenfunction of A and B :
B AAi ,Bj = Bj Ai Ai ,Bj
ABAi ,Bj = Ai Bj Ai ,Bj
Subtracting the equations, we realize a compact notation for dening what
is called a commutator:
[A, B ] = AB B A
The commutator is itself either zero or an operator
The order of operations is important and will give unique commutators
depending on this ordering
For two physical properties to be simultaneously observable, their
operator representations must commute.
Thus,
A Bf (x) B Af (x) = 0
2 operators that commute
Example Problem 17.1: whether Determine the momentum operator commutes with the a) kinetic energy and b) total energy operators.
a). To determine whether the two operators commute (and importantly, to
determine whether the two observables associated with those operators can
be known simultaneously), one considers the following:
2
momentum and kinetic energy
ih
d
dx
h2 d2
2m dx2
f (x)
i
h3 d3
2m dx3
h2 d2
2m dx2
ih
f (x) i
h3 d3
2m dx3
d
f (x)
dx
f (x)
0
Thus, the momentum and kinetic energy operators commute
b). momentum and total energy
ih
d
dx
h2 d2
h2 d2
+ V (x) f (x)
+ V (x)
2m dx2
2m dx2
ih
d
f (x)
dx
(1)
Since the momentum and kinetic energy operators commute from part
a, we can write
d
V (x)f (x)) + ihV (x)
dx
d
d
ihV (x) f (x) ihf (x) V (x) + ihV (x)
dx
dx
d
f (x)
dx
d
f (x)
dx
d
ihf (x) V (x)
dx
ih
(2)
Thus, the commutator for the momentum and total energy reduces as follows:
d
d
d
= V (x), ih
= ih V (x)
H, ih
dx
dx
dx
The last equation does not equal zero identically, and thus we see two
things: 1. the momentum and total energy do not commute 2. the commutator reduces to a unique operation (we will see this again with respect to
angular momentum)
Heisenberg Uncertainty Principle
Recall the discussion of the free particle. For that system, we determined
that the energy (and momentum) spectrum is continuous since there were
no boundary conditions imposed on the wavefunction (thus we arrive at
plan-wave representations of the particle-wave entity).
The important point for the present dicussion is the relation between our
knowledge of the momentum of the particle and its position. We note a few
things:
3
The Heisenberg Uncertainty principle is stated as:
px
h
2
For a quantum mechanical description of a particles dynamics, we
cannot know exactly and simultaneously both the particles position
and momentum. We must accept an uncertainty in measurements of
these quantities as given by the inequality.
To relate the uncertainty principle to variances and statistical measures, the relation:
x p
h
2
can be used in conjunction with wavefunctions and denitions of average properties:
2
p =< p2 > < p >2
4
(likewise f or x )