Canonical
commutation relation

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In quantum mechanics (physics), the
canonical commutation relation is the fundamental relation between canonical conjugate
quantities (quantities which are related by definition such that one is the Fourier transform of
another). For example,

between the position x and momentum
px in the x direction of a point particle in one
dimension, where [x , px] =
x px − px
x is the commutator of x and px , i is the imaginary unit, and ℏ is the reduced Planck's constant
h/2π
.

This relation is attributed to Max Born (1925),[1]
who called it a "quantum condition" serving as a postulate of the theory; it was
noted by E. Kennard
(1927)[2]
to imply the Heisenberg uncertainty
principle.

Contents

1 Relation to classical
mechanics

2 Representations

3 Generalizations

4 Gauge
invariance

5 Angular momentum
operators

6 See also

7 References

Relation to
classical mechanics[edit]

By contrast, in classical physics, all
observables commute and the commutator would be zero.
However, an analogous relation exists, which is obtained by replacing the
commutator with the Poisson bracket
multiplied by iℏ:

This observation led Dirac to propose that the
quantum counterparts
of classical observables f, g satisfy

In 1946, Hip Groenewold
demonstrated that a general systematic correspondence between quantum
commutators and Poisson brackets could not hold consistently. [3]
However, he did appreciate that such a systematic correspondence does, in fact,
exist between the quantum commutator and a deformation of the Poisson
bracket, the Moyal bracket, and, in
general, quantum operators and classical observables and distributions in phase
space. He thus finally elucidated the correspondence mechanism, Weyl quantization,
that underlies an alternate equivalent mathematical approach to quantization
known as deformation
quantization.[3]

Representations[edit]

The group H3(ℝ) generated by exponentiation of the Lie
algebra specified by these commutation relations, [x, p] = iℏ, is called the Heisenberg group.

According to the standard mathematical
formulation of quantum mechanics, quantum observables such as x and p should be
represented as self-adjoint
operators on some Hilbert space. It is
relatively easy to see that two operators
satisfying the above canonical commutation relations cannot both be bounded—try taking the
Trace of both
sides of the relations and use the relation Trace(AB) = Trace(BA); one gets a
finite number on the right and zero on the left.

The canonical commutation relations can be rendered somewhat "tamer" by
writing them in terms of the (bounded) unitary operators exp(i t x) and exp(i s p) which do admit finite-dimensional
representations. The resulting braiding relations for these are the so-called Weyl
relations exp(i t x) exp(i s p) =
exp(−iℏ s t) exp(i s p) exp(i t x).
The corresponding group commutator is then exp(i t x) exp(i s p) exp(−i t x) exp(−i s p)
= exp(−iℏ s t). The uniqueness of the canonical commutation
relations between position and momentum is then guaranteed by the Stone–von
Neumann theorem.

Generalizations[edit]

The simple formula

valid for the quantization of
the simplest classical system, can be generalized to the case of an arbitrary Lagrangian
.[4]
We identify canonical coordinates (such as x in the example above, or a field φ(x) in the case of quantum field
theory) and canonical momenta πx (in the example above it is p, or more generally, some functions involving
the derivatives of the canonical
coordinates with respect to time):

This definition of the canonical momentum ensures that one of the Euler–Lagrange
equations has the form

The canonical commutation relations then amount to

where δij is the Kronecker delta.

Further, it can be easily shown that

Gauge invariance[edit]

Canonical quantization is applied, by definition, on canonical
coordinates. However, in the presence of an electromagnetic
field, the canonical momentum p
is not gauge invariant. The
correct gauge-invariant momentum (or "kinetic momentum") is

(SI units)

(cgs units),

where q is the particle's electric charge, A is the vector potential, and
c is the speed of light. Although
the quantity pkin is the "physical
momentum", in that it is the quantity to be identified with momentum in
laboratory experiments, it does not satisfy the canonical commutation
relations; only the canonical momentum does that. This can be seen as
follows.

The non-relativistic Hamiltonian
for a quantized charged particle of mass m
in a classical electromagnetic field is (in cgs units)

where A is the three-vector potential
and
is the scalar potential. This
form of the Hamiltonian, as well as the Schrödinger
equation ,
the Maxwell equations and
the Lorentz force law are
invariant under the gauge transformation

where

and
is the gauge function.

The angular momentum
operator is

and obeys the canonical quantization relations

defining the Lie algebra for so(3), where
is the Levi-Civita symbol.
Under gauge transformations, the angular momentum transforms as

The gauge-invariant angular momentum (or "kinetic angular momentum") is given
by

which has the commutation relations

where

is the magnetic field. The
inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm
effect.

Angular momentum
operators[edit]

From Lx = y pz
− z py, etc, it follows directly from the above
that

where
is the Levi-Civita symbol
and simply reverses the sign of the answer under pairwise interchange of the
indices. An analogous relation holds for the spin operators.

All such nontrivial commutation relations for pairs of operators lead to
corresponding uncertainty
relations,[5]
involving positive semi-definite expectation contributions by their respective
commutators and anticommutators. In general, for two Hermitian
operators A and B, consider expectation values in a system in
the state ψ, the variances around the
corresponding expectation values being (ΔA)2 ≡ ⟨(A − ⟨A⟩)2⟩, etc.

Then

where [A, B] ≡ A B −
B A is the commutator of
A and B, and {A, B}
≡ A B + B A is the anticommutator.

This follows through use of the Cauchy–Schwarz
inequality, since |⟨A2⟩| |⟨B2⟩| ≥ |⟨A B⟩|2, and A B = ([A, B] +
{A, B})/2 ; and similarly for the shifted operators A − ⟨A⟩ and B − ⟨B⟩. (cf. Uncertainty
principle derivations.)

Judicious choices for A and B yield Heisenberg's familiar uncertainty
relation for x and p, as usual.

Here, for Lx and Ly ,[5]
in angular momentum multiplets ψ = |ℓ,m⟩, one has ⟨Lx2⟩ = ⟨Ly2⟩ = (ℓ (ℓ + 1) −
m2) ℏ2/2 , so the above inequality yields
useful constraints such as a lower bound on the Casimir invariant
ℓ (ℓ + 1) ≥ m (m + 1), and hence
ℓ ≥ m, among
others.

See also[edit]

Canonical
quantization

CCR algebra

Lie derivative

Moyal bracket

References[edit]

^
Born, M.; Jordan, P.
(1925). "Zur Quantenmechanik". Zeitschrift für Physik 34: 858.
doi:10.1007/BF01328531.

^
Kennard, E. H.
(1927). "Zur Quantenmechanik einfacher Bewegungstypen". Zeitschrift für
Physik 44 (4–5): 326–352. doi:10.1007/BF01391200.

^ a
b
Groenewold, H. J.
(1946). "On the principles of elementary quantum mechanics". Physica
12 (7): 405–460. doi:10.1016/S0031-8914(46)80059-4.

^
Townsend, J. S. (2000).
A Modern Approach to Quantum Mechanics. Sausalito, CA: University
Science Books. ISBN 1-891389-13-0.

^ a
b
Robertson, H. P.
(1929). "The Uncertainty Principle". Physical Review
34 (1): 163–164. Bibcode:1929PhRv...34..163R. doi:10.1103/PhysRev.34.163.

Retrieved from "http://en.wikipedia.org/w/index.php?title=Canonical_commutation_relation&oldid=562288033"

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