Canonical commutation relation

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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 p in the x direction of a point particle in one dimension, where [x,px] = xpx − pxx is the so-called commutator of x and px, i is the imaginary unit and \hbar is the reduced Planck's constant h / 2π. This relation is attributed to Max Born, and it implies the Heisenberg uncertainty principle.

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[edit] 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 and the constant i\hbar with 1:

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

This observation led Dirac to postulate that, in general, the quantum counterparts \hat f,\hat g of classical observables f,g should satisfy

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

In 1927, Hermann Weyl showed that a literal correspondence between a quantum operator and a classical distribution in phase space could not hold. However, he did propose a mechanism, Weyl quantization, that underlies a mathematical approach to quantization known as deformation quantization.

[edit] Representations

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. The result is 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 the commutation relations is called the Heisenberg group.

[edit] Generalizations

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}. 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 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 say

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

where δij is the Kronecker delta.


Also, it can be shown,

[F(x),p] = i\hbar\frac{\partial F(x)}{\partial x},

[edit] Gauge invariance

Canonical quantization is performed, 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-eA/c , \,\!

where e is the quantum of electric charge, and A is the vector potential and c is the speed of light. Although the quantity p 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

H=\frac{1}{2m} \left(p-\frac{eA}{c}\right)^2 +e\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}
\psi\to\psi^\prime=U\psi
H\to H^\prime= U HU^\dagger

where

U=\exp \left( \frac{ie\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 {e}{\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{eA}{c}\right)

which has the commutation relations

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

where

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.

[edit] Examples

[edit] Angular momentum operators

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

Here εxyz is the Levi-Civita symbol and simply reverses the sign of the answer under acyclic permutation of the indices. An analogous relation holds for the spin operators.

[edit] See also

[edit] References

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