Canonical commutation relation explained

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 of another). For example,$$[\backslash hat\; x,\backslash hat\; p\_x]\; =\; i\backslash hbar\; \backslash mathbb$$

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, and

I is the unit operator. In general, position and momentum are vectors of operators and their commutation relation between different components of position and momentum can be expressed as$$[\backslash hat\; x\_i,\backslash hat\; p\_j]\; =\; i\backslash hbar\; \backslash delta\_,$$where \delta _{ij} is the Kronecker delta.

This relation is attributed to Werner Heisenberg, Max Born and Pascual Jordan (1925),^{[1] [2]} who called it a "quantum condition" serving as a postulate of the theory; it was noted by E. Kennard (1927)^[3] 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 multiplied by,$$\backslash \; =\; 1\; \backslash ,\; .$$

This observation led Dirac to propose that the quantum counterparts

\hat{f} , of classical observables, satisfy$$[\backslash hat\; f,\backslash hat\; g]=\; i\backslash hbar\backslash widehat\; \backslash ,\; .$$

In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently.^{[4] [5]}

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.^{[4] [6]}

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:$$\backslash begin\; \backslash dot\; =\; \backslash frac\; =\; \backslash ;\; \backslash \backslash \; \backslash dot\; =\; -\backslash frac\; =\; \backslash .\backslash end$$

In quantum mechanics the Hamiltonian

\hat{H} , (generalized) coordinate \hat{Q} and (generalized) momentum \hat{P} are all linear operators.

The time derivative of a quantum state is -

i\hat{H}/\hbar (by Schrödinger equation). Equivalently, since the operators are not explicitly time-dependent, they can be seen to be evolving in time (see Heisenberg picture) according to their commutation relation with the Hamiltonian:$$\backslash frac\; =\; \backslash frac\; [\backslash hat\{H\},\backslash hat\{Q\}]$$$$\backslash frac\; =\; \backslash frac\; [\backslash hat\{H\},\backslash hat\{P\}]\; \backslash ,\backslash ,\; .$$

In order for that to reconcile in the classical limit with Hamilton's equations of motion,

[\hat{H},\hat{Q}] must depend entirely on the appearance of \hat{P} in the Hamiltonian and [\hat{H},\hat{P}] must depend entirely on the appearance of \hat{Q} 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):$$[\backslash hat\{H\},\backslash hat\{Q\}]\; =\; \backslash frac\; \backslash cdot\; [\backslash hat\{P\},\backslash hat\{Q\}]$$$$[\backslash hat\{H\},\backslash hat\{P\}]\; =\; \backslash frac\; \backslash cdot\; [\backslash hat\{Q\},\backslash hat\{P\}]\; \backslash ,\; .$$

In order to obtain the classical limit we must then have$$[\backslash hat\{Q\},\backslash hat\{P\}]\; =\; i\; \backslash hbar\; ~\; \backslash mathbb.$$

Weyl relations

H _{3(R)} generated by exponentiation of the 3-dimensional Lie algebra determined by the commutation relation [\hat{x},\hat{p}] = i\hbar is called the Heisenberg group. This group can be realized as the group of 3 x 3 upper triangular matrices with ones on the diagonal.^[7]

According to the standard mathematical formulation of quantum mechanics, quantum observables such as

\hat{x} and \hat{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. Certainly, if \hat{x} and \hat{p} were trace class operators, the relation \operatorname{Tr}(AB) = \operatorname{Tr}(BA) gives a nonzero number on the right and zero on the left.

Alternately, if

\hat{x} and \hat{p} were bounded operators, note that [\hat{x} ^{n,\hat{p}]=i\hbar} n \hat{x} ^{n-1} , hence the operator norms would satisfy$$2\; \backslash left\backslash |\backslash hat\backslash right\backslash |\; \backslash left\backslash |\backslash hat^\backslash right\backslash |\; \backslash left\backslash |\backslash hat\backslash right\backslash |\; \backslash geq\; n\; \backslash hbar\; \backslash left\backslash |\backslash hat^\backslash right\backslash |,$$ so that, for any n,$$2\; \backslash left\backslash |\backslash hat\backslash right\backslash |\; \backslash left\backslash |\backslash hat\backslash right\backslash |\; \backslash geq\; n\; \backslash hbar$$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

\exp(it\hat{x}) and \exp(is\hat{p}) . The resulting braiding relations for these operators are the so-called Weyl relations$$\backslash exp(it\backslash hat)\backslash exp(is\backslash hat)=\backslash exp(-ist/\backslash hbar)\backslash exp(is\backslash hat)\backslash exp(it\backslash hat).$$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.

For technical reasons, the Weyl relations are not strictly equivalent to the canonical commutation relation

[\hat{x},\hat{p}] = i\hbar . If \hat{x} and \hat{p} 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.^[8] 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.^[9] (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 range over

Z /n , can be realized on a finite-dimensional Hilbert space by means of the clock and shift matrices.

Generalizations

{ l L} .^[10] 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 of the canonical coordinates with respect to time):$$\backslash pi\_i\; \backslash \; \backslash stackrel\backslash \; \backslash frac.$$

This definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the form$$\backslash frac\; \backslash pi\_i\; =\; \backslash frac.$$

The canonical commutation relations then amount to$$[x\_i,\backslash pi\_j]\; =\; i\backslash hbar\backslash delta\_\; \backslash ,$$where is the Kronecker delta.

Further, it can be shown that$$[F(\backslash vec\{x\}),p\_i]\; =\; i\backslash hbar\backslash frac;\; \backslash qquad\; [x\_i,\; F(\backslash vec\{p\})]\; =\; i\backslash hbar\backslash frac.$$

Using

C ^{k} _{n +1} = C ^{k} _{n} + C ^{k -1} _{n} , it can be shown that by mathematical induction$$\backslash left[\backslash hat\{x\}^n,\backslash hat\{p\}^m\backslash right]\; =\; \backslash sum\_^\; =\; \backslash sum\_^,$$generally known as McCoy's formula.^[11]

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

p _{kin} = p - qA (SI units) p _{kin} = p - (qA)/(c) (cgs units),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 in a classical electromagnetic field is (in cgs units)$$H=\backslash frac\; \backslash left(p-\backslash frac\backslash right)^2\; +q\backslash phi$$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 are invariant under the gauge transformation$$A\backslash to\; A\text{'}\; =\; A+\backslash nabla\; \backslash Lambda$$$$\backslash phi\backslash to\; \backslash phi\text{'}\; =\; \backslash phi-\backslash frac\; \backslash frac$$$$\backslash psi\; \backslash to\; \backslash psi\text{'}\; =\; U\backslash psi$$$$H\backslash to\; H\text{'}\; =\; U\; H\; U^\backslash dagger,$$where $$U=\backslash exp\; \backslash left(\backslash frac\backslash right)$$ and is the gauge function.

The angular momentum operator is$$L=r\; \backslash times\; p\; \backslash ,\backslash !$$and obeys the canonical quantization relations$$[L\_i,\; L\_j]=\; i\backslash hbar\; L\_k$$defining the Lie algebra for so(3), where

\epsilon _{ijk} is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as$$\backslash langle\; \backslash psi\; \backslash vert\; L\; \backslash vert\; \backslash psi\; \backslash rangle\; \backslash to\backslash langle\; \backslash psi^\backslash prime\; \backslash vert\; L^\backslash prime\; \backslash vert\; \backslash psi^\backslash prime\; \backslash rangle\; =\backslash langle\; \backslash psi\; \backslash vert\; L\; \backslash vert\; \backslash psi\; \backslash rangle\; +\backslash frac\; \backslash langle\; \backslash psi\; \backslash vert\; r\; \backslash times\; \backslash nabla\; \backslash Lambda\; \backslash vert\; \backslash psi\; \backslash rangle\; \backslash ,\; .$$

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by$$K=r\; \backslash times\; \backslash left(p-\backslash frac\backslash right),$$which has the commutation relations$$[K\_i,K\_j]=i\backslash hbar\; ^\backslash left(K\_k+\backslash frac\; x\_k\backslash left(x\; \backslash cdot\; B\backslash right)\backslash right)$$where $$B=\backslash nabla\; \backslash times\; A$$ 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,^[12] 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.

Then$$\backslash Delta\; A\; \backslash ,\; \backslash Delta\; B\; \backslash geq\; \backslash frac\; \backslash sqrt,$$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, etc., one has that$$[\{L\_x\},\; \{L\_y\}]\; =\; i\; \backslash hbar\; \backslash epsilon\_,$$where

\epsilon _{xyz} 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 .

Consequently, the above inequality applied to this commutation relation specifies$$\backslash Delta\; L\_x\; \backslash Delta\; L\_y\; \backslash geq\; \backslash frac\; \backslash sqrt~,$$hence$$\backslash sqrt$$

\geq \frac \vert m\vertand therefore$$\backslash ell(\backslash ell+1)-m^2\backslash geq\; |m|\; ~,$$so, then, it yields useful constraints such as a lower bound on the Casimir invariant:, and hence, among others.

See also

• Canonical quantization
• CCR and CAR algebras
• Conformastatic spacetimes
• Lie derivative
• Moyal bracket
• Stone–von Neumann theorem

References

• .
• .

Notes and References

1. Web site: The Development of Quantum Mechanics.
2. Born . M. . Jordan . P. . 10.1007/BF01328531 . Zur Quantenmechanik . Zeitschrift für Physik . 34 . 858–888 . 1925 . 1 . 1925ZPhy...34..858B . 186114542 .
3. Kennard . E. H. . Zur Quantenmechanik einfacher Bewegungstypen . 10.1007/BF01391200 . Zeitschrift für Physik . 44 . 4–5 . 326–352 . 1927 . 1927ZPhy...44..326K . 121626384 .
4. Groenewold . H. J. . On the principles of elementary quantum mechanics . 10.1016/S0031-8914(46)80059-4 . Physica . 12 . 7 . 405–460 . 1946 . 1946Phy....12..405G .
5. Theorem 13.13
6. Curtright . T. L. . Zachos . C. K. . 10.1142/S2251158X12000069 . Quantum Mechanics in Phase Space . Asia Pacific Physics Newsletter . 01 . 37–46 . 2012 . 1104.5269 . 119230734 .
7. Section 1.2.6 and Proposition 3.26
8. See Section 5.2 of for an elementary derivation
9. Example 14.5
10. Book: Townsend, J. S. . A Modern Approach to Quantum Mechanics . registration . University Science Books . Sausalito, CA . 2000 . 1-891389-13-0 .
11. McCoy, N. H. (1929), "On commutation formulas in the algebra of quantum mechanics", Transactions of the American Mathematical Society 31 (4), 793-806 online
12. H. P. . Robertson . The Uncertainty Principle . . 34 . 1 . 1929 . 163–164 . 10.1103/PhysRev.34.163 . 1929PhRv...34..163R .