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\hbar

between the position Template:Mvar and momentum Template:Mvar in the Template:Mvar direction of a point particle in one dimension, where [x , px] = x pxpx x is the commutator of Template:Mvar and Template:Mvar, Template:Mvar is the imaginary unit, and 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.

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:

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

This observation led Dirac to propose that the quantum counterparts Template:Mvar, Template:Mvar of classical observables Template:Mvar, Template:Mvar 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. [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 H3(ℝ) generated by exponentiation of the Lie algebra specified by these commutation relations, [x, p] = i, 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 tx) and exp(i sp), which do admit finite-dimensional representations. The resulting braiding relations for these are the so-called Weyl relations

exp(i tx) exp(i sp) = exp(−iℏ s t) exp(i sp) exp(i tx).

The corresponding group commutator is then

exp(i tx) exp(i sp) exp(−i tx) exp(−i sp) = exp(−iℏ 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\hbar, \,

valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian {\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 πx (in the example above it is Template:Mvar, 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

[p_i,F(\vec{x})] = -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 Template:Mvar 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 Template:Mvar is the particle's electric charge, Template:Mvar 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 Template:Mvar in a classical electromagnetic field is (in cgs units)

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

where Template:Mvar is the three-vector potential and \phi is the scalar potential. This form of the Hamiltonian, as well as the Schrödinger 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{iq\Lambda}{\hbar c}\right)

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

The angular momentum operator 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 \epsilon_{ijk} 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)

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.

Angular momentum operators

From Lx = y pzz py, etc, it follows directly from the above that

= 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,[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 A)2 ≡ (AA)2, etc.

Then

\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] ≡ A BB 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 |A2| |B2| ≥ |A B|2, and A B = ([A, B] + {A, B})/2 ; and similarly for the shifted operators AA and BB. (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 Lx2 = Ly2 = (Template:Ell (Template:Ell + 1) − m2) ℏ2/2 , 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:Ellm, among others.

See also

References

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