Wikipedia / Canonical commutation relation
Wikipedia / Canonical commutation relation
In quantum mechanics, 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
between the position operator and momentum operator in the direction of a point particle in one dimension, where is the commutator of and , is the imaginary unit, and is the reduced Planck constant ( is the Kronecker delta.
This relation is attributed to Werner Heisenberg, Max Born and Pascual Jordan (1925), who called it a "quantum condition" serving as a postulate of the theory; it was noted by E. Kennard (1927) to imply the Heisenberg uncertainty principle. The Stone–von Neumann theorem gives a uniqueness result for operators satisfying (an exponentiated form of) the canonical commutation relation.
Relation to classical mechanics
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
This observation led Dirac of classical observables ,
In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.
However, he further appreciated that such a systematic correspondence does, in fact, exist between the quantum commutator and a deformation of the Poisson bracket, today called the Moyal bracket, and, in general, quantum operators and classical observables and distributions in phase space. He thus finally elucidated the consistent correspondence mechanism, the Wigner–Weyl transform, that underlies an alternate equivalent mathematical representation of quantum mechanics known as deformation quantization.
Derivation from Hamiltonian mechanics
According to the correspondence principle, in certain limits the quantum equations of states must approach Hamilton's equations of motion. The latter state the following relation between the generalized coordinate q (e.g. position) and the generalized momentum p
are all linear operators.
The time derivative of a quantum state (by the Schrödinger equation). Equivalently, since in the Schrödinger picture the operators are not explicitly time-dependent, the operators can be seen to be evolving in time (for a contrary perspective where the operators are time dependent, see Heisenberg picture
in the Hamiltonian. Further, since the Hamiltonian operator depends on the (generalized) coordinate and momentum operators, it can be viewed as a functional, and we may write (using functional derivatives
Weyl relations
The group generated by exponentiation of the 3-dimensional Lie algebra is called the Heisenberg group upper triangular matrices with ones on the diagonal.
According to the standard mathematical formulation of quantum mechanics should be represented as self-adjoint operators on some Hilbert space. It is relatively easy to see that two operatorss satisfying the above canonical commutation relations cannot both be bounded were trace class gives a nonzero number on the right and zero on the left.
so that, for any n However, can be arbitrarily large, so at least one operator cannot be bounded, and the dimension of the underlying Hilbert space cannot be finite. If the operators satisfy the Weyl relations (an exponentiated version of the canonical commutation relations, described below) then as a consequence of the Stone–von Neumann theorem, both operators must be unbounded.
Still, these canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operators. The resulting braiding relations for these operators are the so-called Weyl relations These relations may be thought of as an exponentiated version of the canonical commutation relations; they reflect that translations in position and translations in momentum do not commute. One can easily reformulate the Weyl relations in terms of the representations of the Heisenberg group.
The uniqueness of the canonical commutation relations—in the form of the Weyl relations—is then guaranteed by the Stone–von Neumann theorem.
were bounded operators, then a special case of the Baker–Campbell–Hausdorff formula would allow one to "exponentiate" the canonical commutation relations to the Weyl relations. Since, as we have noted, any operators satisfying the canonical commutation relations must be unbounded, the Baker–Campbell–Hausdorff formula does not apply without additional domain assumptions. Indeed, counterexamples exist satisfying the canonical commutation relations but not the Weyl relations. (These same operators give a counterexample to the naive form of the uncertainty principle.) These technical issues are the reason that the Stone–von Neumann theorem is formulated in terms of the Weyl relations.
A discrete version of the Weyl relations, in which the parameters s and t, can be realized on a finite-dimensional Hilbert space by means of the clock and shift matrices.
Generalizations
, it can be shown that by mathematical induction generally known as McCoy's formula.
valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian. We identify canonical coordinates (such as in the example above, or a field in the case of quantum field theory) and canonical momenta (in the example above it is , or more generally, some functions involving the derivatives
This definition of the canonical momentum ensures that one of the Euler–Lagrange equations
where is the Kronecker delta.
Gauge invariance
Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is
where is the particle's electric charge, is the vector potential, and is the speed of light. Although the quantity 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 where 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 and is the gauge function.
The angular momentum operator defining the Lie algebra for so(3) is the Levi-Civita symbol
is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect.
Uncertainty relation and commutators
All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations, involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators and , consider expectation values in a system in the state , the variances around the corresponding expectation values being , etc.
where is the commutator of and , and is the anticommutator.
This follows through use of the Cauchy–Schwarz inequality, since , and ; and similarly for the shifted operators and . (Cf. uncertainty principle derivations.)
Substituting for and (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation for and , as usual.
Uncertainty relation for angular momentum operators
For the angular momentum operators 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.
Here, for and , in angular momentum multiplets , one has, for the transverse components of the Casimir invariant , the -symmetric relations
,
as well as .
so, then, it yields useful constraints such as a lower bound on the Casimir invariant: , and hence , among others.
See also
References
.
.