### Canonical Momentum

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,

- $[x,p\_x]\; =\; i\backslash hbar$

between the position Template:Mvar and momentum Template:Mvar in the Template:Mvar direction of a point particle in one dimension, where [*x* , *p*_{x}] = *x* *p*_{x} − *p*_{x} *x* is the commutator of Template:Mvar and Template:Mvar, Template:Mvar is the imaginary unit, and ℏ is the reduced Planck's constant Template:Sfrac .

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.

## Contents

## 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:

- $\backslash \{x,p\backslash \}\; =\; 1\; \backslash ,\; .$

This observation led Dirac to propose that the quantum counterparts Template:Mvar, Template:Mvar of classical observables Template:Mvar, Template:Mvar satisfy

- $[\backslash hat\; f,\backslash hat\; g]=\; i\backslash hbar\backslash widehat\{\backslash \{f,g\backslash \}\}\; \backslash ,\; .$

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, Weyl quantization, that underlies an alternate equivalent mathematical approach to quantization known as deformation quantization.^{[3]}

## Representations

The group *H*_{3}(ℝ) generated by exponentiation of the Lie algebra specified by these commutation relations, [**x**, **p**] = *i*ℏ, 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.^{[4]}

These canonical commutation relations can be rendered somewhat "tamer" by writing them in terms of the (bounded) unitary operators exp(*i t* **x**) and exp(*i s* **p**), which do admit finite-dimensional representations. The resulting braiding relations for these are the so-called Weyl relations

- exp(
*i t***x**) exp(*i s***p**) = exp(−*i*ℏ*s t*) exp(*i s***p**) exp(*i t***x**).

The corresponding group commutator is then

- exp(
*i t***x**) exp(*i s***p**) exp(−*i t***x**) exp(−*i s***p**) = exp(−*i*ℏ*s t*).

The uniqueness of the canonical commutation relations between position and momentum is then guaranteed by the Stone–von Neumann theorem.

## Generalizations

The simple formula

- $[x,p]\; =\; i\backslash hbar,\; \backslash ,$

valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian $\{\backslash mathcal\; L\}$.^{[5]} We identify **canonical coordinates** (such as Template:Mvar in the example above, or a field φ(*x*) in the case of quantum field theory) and **canonical momenta** π_{x} (in the example above it is Template:Mvar, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time):

- $\backslash pi\_i\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\}\{\backslash partial(\backslash partial\; x\_i\; /\; \backslash partial\; t)\}.$

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

- $\backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; \backslash pi\_i\; =\; \backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\}\{\backslash partial\; x\_i\}.$

The canonical commutation relations then amount to

- $[x\_i,\backslash pi\_j]\; =\; i\backslash hbar\backslash delta\_\{ij\},\; \backslash ,$

where *δ*_{ij} is the Kronecker delta.

Further, it can be easily shown that

- $[p\_i,F(\backslash vec\{x\})]\; =\; -i\backslash hbar\backslash frac\{\backslash partial\; F(\backslash vec\{x\})\}\{\backslash partial\; x\_i\};\; \backslash qquad\; [x\_i,\; F(\backslash vec\{p\})]\; =\; i\backslash hbar\backslash frac\{\backslash partial\; F(\backslash vec\{p\})\}\{\backslash partial\; p\_i\}.$

## Gauge invariance

Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum Template:Mvar is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is

- $p\_\backslash textrm\{kin\}\; =\; p\; -\; qA\; \backslash ,\backslash !$ (SI units) $p\_\backslash textrm\{kin\}\; =\; p\; -\; \backslash frac\{qA\}\{c\}\; \backslash ,\backslash !$ (cgs units),

where Template:Mvar is the particle's electric charge, Template:Mvar is the vector potential, and *c* is the speed of light. Although the quantity *p*_{kin} 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 Template:Mvar in a classical electromagnetic field is (in cgs units)

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

where Template:Mvar is the three-vector potential and $\backslash phi$ is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger equation $H\backslash psi=i\backslash hbar\; \backslash partial\backslash psi/\backslash partial\; t$, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

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

where

- $U=\backslash exp\; \backslash left(\; \backslash frac\{iq\backslash Lambda\}\{\backslash hbar\; c\}\backslash right)$

and $\backslash Lambda=\backslash Lambda(x,t)$ is the gauge function.

The angular momentum operator is

- $L=r\; \backslash times\; p\; \backslash ,\backslash !$

and obeys the canonical quantization relations

- $[L\_i,\; L\_j]=\; i\backslash hbar\; \{\backslash epsilon\_\{ijk\}\}\; L\_k$

defining the Lie algebra for so(3), where $\backslash epsilon\_\{ijk\}$ is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as

- $\backslash langle\; \backslash psi\; \backslash vert\; L\; \backslash vert\; \backslash psi\; \backslash rangle\; \backslash to$

\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

- $K=r\; \backslash times\; \backslash left(p-\backslash frac\{qA\}\{c\}\backslash right),$

which has the commutation relations

- $[K\_i,K\_j]=i\backslash hbar\; \{\backslash epsilon\_\{ij\}\}^\{\backslash ,k\}$

\left(K_k+\frac{q\hbar}{c} x_k \left(x \cdot B\right)\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.

## Angular momentum operators

From *L*_{x} = *y p _{z}* −

*z p*, etc, it follows directly from the above that

_{y}- $=\; i\; \backslash hbar\; \backslash epsilon\_\{xyz\}\; \{L\_z\},$

where $\backslash epsilon\_\{xyz\}$ 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,^{[6]} involving positive semi-definite expectation contributions by their respective commutators and anticommutators. In general, for two Hermitian operators Template:Mvar and Template:Mvar, consider expectation values in a system in the state Template:Mvar, the variances around the corresponding expectation values being (Δ*A*)^{2 ≡ ⟨(A − ⟨A⟩)2⟩}, etc.

Then

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

where [*A*, *B*] ≡ *A B* − *B A* is the commutator of Template:Mvar and Template:Mvar, and {*A*, *B*} ≡ *A B* + *B A* is the anticommutator.

This follows through use of the Cauchy–Schwarz inequality, since
|⟨*A*^{2}⟩| |⟨*B*^{2}⟩| ≥ |⟨*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.)

Judicious choices for Template:Mvar and Template:Mvar yield Heisenberg's familiar uncertainty relation for Template:Mvar and Template:Mvar, as usual.

Here, for Template:Mvar and Template:Mvar,^{[6]} in angular momentum multiplets *ψ* = |*Template:Ell*,*m*⟩, one has ⟨*L _{x}*

^{2}⟩ = ⟨

*L*

_{y}^{2}⟩ = (

*Template:Ell*(

*Template:Ell*+ 1) −

*m*

^{2}) ℏ

^{2}/2 , so the above inequality yields useful constraints such as a lower bound on the Casimir invariant

*Template:Ell*(

*Template:Ell*+ 1) ≥

*m*(

*m*+ 1), and hence

*Template:Ell*≥

*m*, among others.