Canonical commutation relation

Canonical commutation relation

In physics, the canonical commutation relation is the relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another), for example:

[x,p_x] = i\hbar

between the position x and momentum px in the x direction of a point particle in one dimension, where [x,px] = xpxpxx 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, and it was noted by E. Kennard (1927) to imply the Heisenberg uncertainty principle.


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 i ħ:

\{x,p\} = 1 \,      .

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

[\hat f,\hat g]= i\hbar\widehat{\{f,g\}} \, .

In 1946, Hip Groenewold demonstrated that a general systematic correspondence between quantum commutators and Poisson brackets could not hold consistently. 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.


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 canonical commutation relations cannot both be bounded. The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operators e ikx and e iap, which admit finite-dimensional representations as well. The resulting braiding relations for these are the so-called Weyl relations. The uniqueness of the canonical commutation relations between position and momentum is guaranteed by the Stone-von Neumann theorem. The group associated with these commutation relations is called the Heisenberg group.


The simple formula

[x,p] = i\hbar,

valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian {\mathcal L}.[1] 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):

\pi_i \ \stackrel{\mathrm{def}}{=}\  \frac{\partial {\mathcal L}}{\partial(\partial x_i / \partial t)}.

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

\frac{\partial}{\partial t} \pi_i = \frac{\partial {\mathcal L}}{\partial x_i}.

The canonical commutation relations then amount to

[x_i,\pi_j] = i\hbar\delta_{ij},

where δij is the Kronecker delta.

Further, it can be easily shown that

[F(\vec{x}),p_i] = i\hbar\frac{\partial F(\vec{x})}{\partial x_i}; \qquad [x_i, F(\vec{p})] = i\hbar\frac{\partial F(\vec{p})}{\partial p_i}.

Gauge invariance

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

p_\textrm{kin} = p - qA  \,\!   (SI units)      p_\textrm{kin} = p - \frac{qA}{c} \,\!   (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)

H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi

where A is the three-vector potential and ϕ is the scalar potential. This form of the Hamiltonian, as well as the Schroedinger equation H\psi=i\hbar \partial\psi/\partial t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

A\to A^\prime=A+\nabla \Lambda
\phi\to \phi^\prime=\phi-\frac{1}{c} \frac{\partial \Lambda}{\partial t}
H\to H^\prime= U HU^\dagger ,


U=\exp \left( \frac{iq\Lambda}{\hbar c}\right)

and Λ = Λ(x,t) is the gauge function.

The canonical angular momentum is

L=r \times p \,\!

and obeys the canonical quantization relations

[L_i, L_j]= i\hbar {\epsilon_{ijk}} L_k

defining the Lie algebra for so(3), where εijk 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 {q}{\hbar c}  \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{qA}{c}\right),

which has the commutation relations

[K_i,K_j]=i\hbar {\epsilon_{ij}}^{\,k}
\left(K_k+\frac{q\hbar}{c} x_k 
\left(x \cdot B\right)\right)


B=\nabla \times A

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

Angular momentum operators

 [{L_x}, {L_y}] = i \hbar \epsilon_{xyz} {L_z},

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.

All such nontrivial commutation relations for pairs of operators lead to corresponding uncertainty relations,[2] 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.


 \Delta  A \, \Delta  B \geq  \frac{1}{2} \sqrt{ \left|\left\langle\left[{A},{B}\right]\right\rangle \right|^2 + \left|\left\langle\left\{ A-\langle A\rangle ,B-\langle B\rangle  \right\} \right\rangle \right|^2} ,

where [A,B] ≡ ABBA is the commutator of A and B, and {A,B} ≡ AB+BA is the anticommutator. This follows through use of the Cauchy–Schwarz inequality, since |〈A2〉| |〈B2〉| ≥ |〈AB〉|2, and AB = ([A,B] + {A,B}) /2 ; and similarly for the shifted operators A−〈A〉 and B−〈B〉 . Judicious choices for A and B yield Heisenberg's familiar uncertainty relation, for x and p, as usual; or, here, Lx and Ly , in angular momentum multiplets, ψ = |l , m 〉 , useful constraints such as l (l+1) ≥ m (m+1), and hence lm, among others.

See also


  1. ^ Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. Sausalito, CA: University Science Books. ISBN 1891389130. 
  2. ^ Robertson, H. P. (1929). "The Uncertainty Principle". Physical Review 34 (1): 163–164. Bibcode 1929PhRv...34..163R. doi:10.1103/PhysRev.34.163. 

Wikimedia Foundation. 2010.

Поможем написать курсовую

Look at other dictionaries:

  • Path integral formulation — This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see line integral. The path integral formulation of quantum mechanics is a description of quantum theory which… …   Wikipedia

  • Uncertainty principle — In quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the momentum of the particle uncertain; and conversely, that measuring the momentum of a particle precisely makes the… …   Wikipedia

  • Bosonic field — In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeyed …   Wikipedia

  • Bogoliubov transformation — In theoretical physics, the Bogoliubov transformation, named after Nikolay Bogolyubov, is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into… …   Wikipedia

  • Laplace–Runge–Lenz vector — Throughout this article, vectors and their magnitudes are indicated by boldface and italic type, respectively; for example, left| mathbf{A} ight| = A. In classical mechanics, the Laplace–Runge–Lenz vector (or simply the LRL vector) is a vector… …   Wikipedia

  • Momentum operator — See also: Momentum In quantum mechanics, momentum is defined as an operator on the wave function. The Heisenberg uncertainty principle defines limits on how accurately the momentum and position of a single observable system can be known at once.… …   Wikipedia

  • Commutator — This article is about the mathematical concept. For the relation between canonical conjugate entities, see canonical commutation relation. For the type of electrical switch, see commutator (electric). In mathematics, the commutator gives an… …   Wikipedia

  • Weyl quantization — In mathematics and physics, in the area of quantum mechanics, Weyl quantization is a method for systematically associating a quantum mechanical Hermitian operator with a classical kernel function in phase space invertibly. A synonym is phase… …   Wikipedia

  • Quantum harmonic oscillator — The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because an arbitrary potential can be approximated as a harmonic potential …   Wikipedia

  • CCR and CAR algebras — In quantum field theory, if V is a real vector space equipped with a nonsingular real antisymmetric bilinear form (,) (i.e. a symplectic vector space), the unital * algebra generated by elements of V subject to the relations :fg gf=i(f,g):f*=ffor …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”