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 operator x and momentum operator px in the x direction of a point particle in one dimension, where [x , px] = x pxpx x is the commutator of x and px, i is the imaginary unit, and is the reduced Planck's constant h/2π . In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as

where is the Kronecker delta.

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.

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 i:

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, the Wigner–Weyl transform, that underlies an alternate equivalent mathematical representation of quantum mechanics known as deformation quantization.[3]

Representations[edit]

The group H3(ℝ) generated by exponentiation of the 3-dimensional Lie algebra determined by the commutation relation [x, p] = i 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(A B ) = Trace(B A ); one gets a finite number on the right and zero on the left.

Alternately, note that [xn, p] = i ℏ n xn − 1, hence the operator norms satisfy

2 ‖ p ‖ ‖ x ‖nn ℏ ‖ x ‖n − 1,   so that, for any n,
2 ‖ p ‖ ‖ x ‖ ≥ n ℏ.

However, n can be arbitrarily large, so at least one operator cannot be bounded, and the dimension of the underlying Hilbert space cannot be finite. Utilizing the Weyl relations, below, it can actually be shown that both operators are unbounded.

Still, these canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operators exp(i tx) and exp(i sp), which do admit finite-dimensional representations — cf. Generalizations of Pauli matrices#Construction: The clock and shift matrices.

The resulting braiding relations for these are the so-called Weyl relations

exp(i tx) exp(i sp) = exp(−iℏ s t) exp(i sp) exp(i tx).

The corresponding group commutator is then

exp(i tx) exp(i sp) exp(−i tx) exp(−i sp) = 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 φ is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation = iħ∂ψ/∂t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

where

and Λ=Λ(x,t) 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.

Uncertainty relation and commutators[edit]

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 A)2 ≡ ⟨(A − ⟨A⟩)2, etc.

Then

where [A, B] ≡ A BB 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.)

Substituting for A and B (and taking care with the analysis) yield Heisenberg's familiar uncertainty relation for x and p, as usual.

Uncertainty relation for angular momentum operators[edit]

For the angular momentum operators Lx = y pzz py, etc., one has 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.

Here, for Lx and Ly,[5] in angular momentum multiplets ψ = |,m, one has, for the transverse components of the Casimir invariant Lx2 + Ly2+ Lz2, the z-symmetric relations

Lx2⟩ = ⟨Ly2⟩ = ( ( + 1) − m2) ℏ2/2 ,

as well as Lx⟩ = ⟨Ly⟩ = 0 .

Consequently, the above inequality applied to this commutation relation specifies

hence

and therefore

so, then, it yields useful constraints such as a lower bound on the Casimir invariant:  ( + 1) ≥ m (m + 1), and hence m, among others.

See also[edit]

References[edit]

  1. ^ Born, M.; Jordan, P. (1925). "Zur Quantenmechanik". Zeitschrift für Physik. 34: 858. Bibcode:1925ZPhy...34..858B. doi:10.1007/BF01328531. 
  2. ^ Kennard, E. H. (1927). "Zur Quantenmechanik einfacher Bewegungstypen". Zeitschrift für Physik. 44 (4–5): 326–352. Bibcode:1927ZPhy...44..326K. doi:10.1007/BF01391200. 
  3. ^ a b Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4. 
  4. ^ Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. Sausalito, CA: University Science Books. ISBN 1-891389-13-0. 
  5. ^ 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.