Canonical commutation relation
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In
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
, 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 In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the tr ...
of another). For example, hat x,\hat p_x= i\hbar \mathbb between the position operator and momentum operator in the direction of a point particle in one dimension, where is the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of and , is the
imaginary unit The imaginary unit or unit imaginary number () is a mathematical constant that is a solution to the quadratic equation Although there is no real number with this property, can be used to extend the real numbers to what are called complex num ...
, and is the
reduced Planck constant The Planck constant, or Planck's constant, denoted by h, is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a ...
, and \mathbb 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 hat x_i,\hat p_j= i\hbar \delta_, where \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
. This relation is attributed to
Werner Heisenberg Werner Karl Heisenberg (; ; 5 December 1901 – 1 February 1976) was a German theoretical physicist, one of the main pioneers of the theory of quantum mechanics and a principal scientist in the German nuclear program during World War II. He pub ...
,
Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German-British theoretical physicist who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics, and supervised the work of a ...
and
Pascual Jordan Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matri ...
(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 uncertainty principle, also known as Heisenberg's indeterminacy principle, is a fundamental concept in quantum mechanics. It states that there is a limit to the precision with which certain pairs of physical properties, such as position a ...
. 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 Classical physics refers to physics theories that are non-quantum or both non-quantum and non-relativistic, depending on the context. In historical discussions, ''classical physics'' refers to pre-1900 physics, while '' modern physics'' refers to ...
, all observables commute and the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
would be zero. However, an analogous relation exists, which is obtained by replacing the commutator with the
Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...
multiplied by i\hbar, \ = 1 \, . This observation led Dirac to propose that the quantum counterparts \hat, \hat of classical observables , satisfy hat f,\hat g i\hbar\widehat \, . 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 The phase space of a physical system is the set of all possible physical states of the system when described by a given parameterization. Each possible state corresponds uniquely to a point in the phase space. For mechanical systems, the p ...
. 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 physics, a correspondence principle is any one of several premises or assertions about the relationship between classical and quantum mechanics. The physicist Niels Bohr coined the term in 1920 during the early development of quantum theory; ...
, 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'': \begin \dot = \frac = \; \\ \dot = -\frac = \. \end In quantum mechanics the Hamiltonian \hat, (generalized) coordinate \hat and (generalized) momentum \hat are all linear operators. The time derivative of a quantum state is represented by the operator -i\hat/\hbar (by the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
). 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 physics, the Heisenberg picture or Heisenberg representation is a Dynamical pictures, formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which observables incorporate a dependency on time, but the quantum state, st ...
) according to their commutation relation with the Hamiltonian: \frac = \frac hat,\hat/math> \frac = \frac hat,\hat\,\, . In order for that to reconcile in the classical limit with Hamilton's equations of motion, hat,\hat/math> must depend entirely on the appearance of \hat in the Hamiltonian and hat,\hat/math> must depend entirely on the appearance of \hat 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): hat,\hat= \frac \cdot hat,\hat/math> hat,\hat= \frac \cdot hat,\hat\, . In order to obtain the classical limit we must then have hat,\hat= i \hbar ~ I.


Weyl relations

The group H_3(\mathbb) generated by
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
of the 3-dimensional
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
determined by the commutation relation hat,\hati\hbar is called the Heisenberg group. This group can be realized as the group of 3\times 3 upper triangular matrices with ones on the diagonal. According to the standard
mathematical formulation of quantum mechanics The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, whic ...
, quantum observables such as \hat and \hat should be represented as self-adjoint operators on some
Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...
. It is relatively easy to see that two operators satisfying the above canonical commutation relations cannot both be bounded. Certainly, if \hat and \hat were trace class operators, the relation \operatorname(AB)=\operatorname(BA) gives a nonzero number on the right and zero on the left. Alternately, if \hat and \hat were bounded operators, note that hat^n,\hati\hbar n \hat^, hence the operator norms would satisfy 2 \left\, \hat\right\, \left\, \hat^\right\, \left\, \hat\right\, \geq n \hbar \left\, \hat^\right\, , so that, for any ''n'', 2 \left\, \hat\right\, \left\, \hat\right\, \geq n \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) and \exp(is\hat). The resulting braiding relations for these operators are the so-called Weyl relations \exp(it\hat)\exp(is\hat)=\exp(-ist\hbar)\exp(is\hat)\exp(it\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,\hati\hbar. If \hat and \hat were bounded operators, then a special case of the
Baker–Campbell–Hausdorff formula In mathematics, the Baker–Campbell–Hausdorff formula gives the value of Z that solves the equation e^X e^Y = e^Z for possibly noncommutative and in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultima ...
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 A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a c ...
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 \mathbb/n, can be realized on a finite-dimensional Hilbert space by means of the clock and shift matrices.


Generalizations

It can be shown that (\vec),p_i= i\hbar\frac; \qquad _i, F(\vec)= i\hbar\frac. Using C_^ = C_^ + C_^, it can be shown that by
mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots  all hold. This is done by first proving a ...
\left hat^n,\hat^m\right= \sum_^ = \sum_^ , generally known as McCoy's formula.McCoy, N. H. (1929), "On commutation formulas in the algebra of quantum mechanics", ''Transactions of the American Mathematical Society'' ''31'' (4), 793-80
online
/ref> In addition, the simple formula ,p= i\hbar \, \mathbb ~, 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 In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
) and canonical momenta (in the example above it is , or more generally, some functions involving the
derivative In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is t ...
s of the canonical coordinates with respect to time): \pi_i \ \stackrel\ \frac. This definition of the canonical momentum ensures that one of the Euler–Lagrange equations has the form \frac \pi_i = \frac. The canonical commutation relations then amount to _i,\pi_j= i\hbar\delta_ \, where is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
.


Gauge invariance

Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an
electromagnetic field An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated by and acting upon electric charges. The field at any point in space and time can be regarde ...
, the canonical momentum is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is : p_\text = p - qA \,\! (
SI units The International System of Units, internationally known by the abbreviation SI (from French ), is the modern form of the metric system and the world's most widely used system of measurement. It is the only system of measurement with official st ...
) p_\text = p - \frac \,\! ( cgs units), where is the particle's
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
, is the vector potential, and is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
. 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 Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
for a quantized charged particle of mass in a classical electromagnetic field is (in cgs units) H=\frac \left(p-\frac\right)^2 +q\phi where is the three-vector potential and is the scalar potential. This form of the Hamiltonian, as well as the
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation A\to A' = A+\nabla \Lambda \phi\to \phi' = \phi-\frac \frac \psi \to \psi' = U\psi H\to H' = U H U^\dagger, where U=\exp \left( \frac\right) and is the gauge function. The angular momentum operator is L=r \times p \,\! and obeys the canonical quantization relations _i, L_j i\hbar L_k defining the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
for
so(3) In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a ...
, where \epsilon_ is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as \langle \psi \vert L \vert \psi \rangle \to \langle \psi^\prime \vert L^\prime \vert \psi^\prime \rangle = \langle \psi \vert L \vert \psi \rangle + \frac \langle \psi \vert r \times \nabla \Lambda \vert \psi \rangle \, . The gauge-invariant angular momentum (or "kinetic angular momentum") is given by K=r \times \left(p-\frac\right), which has the commutation relations _i,K_ji\hbar ^ \left(K_k+\frac x_k \left(x \cdot B\right)\right) where B=\nabla \times A is the
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
. 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. Then \Delta A \, \Delta B \geq \frac \sqrt , where is the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
of and , and is the
anticommutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, ...
. This follows through use of the
Cauchy–Schwarz inequality The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is ...
, 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 = i \hbar \epsilon_ , where \epsilon_ 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 \Delta L_x \, \Delta L_y \geq \frac \sqrt~, hence \sqrt \geq \frac \vert m\vert and therefore \ell(\ell+1)-m^2\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

* . * . {{Authority control Quantum mechanics Mathematical physics zh:對易關係