Canonical commutation relation

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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 $x$ and momentum $p x$ in the $x$ direction of a point particle in one dimension, where $[x, p x] = x p x - p x x$ is the commutator of $x$ and $p x$ , $i$ is the imaginary unit, and $ℏ$ is the reduced Planck's constant $h / 2 π$ .

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 $i ℏ$ :

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

This observation led Dirac to propose that the quantum counterparts $f ̂$ , $g ̂$ 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. ^[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

$[x,p]=i\hbar ,\,$

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 $x$ 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 $p$ , 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 $δ ij$ 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 $p$ 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 $p kin$ 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 $m$ 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 $\phi$ 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

$A\to A^{\prime }=A+\nabla \Lambda$

$\phi \to \phi ^{\prime }=\phi -{\frac {1}{c}}{\frac {\partial \Lambda }{\partial t}}$

$\psi \to \psi ^{\prime }=U\psi$

$H\to H^{\prime }=UHU^{\dagger },$

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 $L x = y p z - z p y$ , etc, it follows directly from the above that

$[{L_{x}},{L_{y}}]=i\hbar \epsilon _{{xyz}}{L_{z}},$

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,^[6] 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 $(Δ A) 2 \equiv ⟨ (A - ⟨ A ⟩) 2 ⟩$ , 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 $[A, B] \equiv A B - B A$ is the commutator of $A$ and $B$ , and ${A, B} \equiv A B + B A$ is the anticommutator.

This follows through use of the Cauchy–Schwarz inequality, since $| ⟨ A 2 ⟩ | | ⟨ B 2 ⟩ | \geq | ⟨ A B ⟩ | 2$ , 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 $A$ and $B$ yield Heisenberg's familiar uncertainty relation for $x$ and $p$ , as usual.

Here, for $L x$ and $L y$ ,^[6] in angular momentum multiplets $ψ = | ℓ, m ⟩$ , one has $⟨ L x 2 ⟩ = ⟨ L y 2 ⟩ = (ℓ (ℓ + 1) - m 2) ℏ 2 /2$ , so the above inequality yields useful constraints such as a lower bound on the Casimir invariant $ℓ (ℓ + 1) \geq m (m + 1)$ , and hence $ℓ \geq m$ , among others.

References

↑ Born, M.; Jordan, P. (1925). "Zur Quantenmechanik". Zeitschrift für Physik 34: 858. doi:10.1007/BF01328531.
↑ Kennard, E. H. (1927). "Zur Quantenmechanik einfacher Bewegungstypen". Zeitschrift für Physik 44 (4–5): 326–352. doi:10.1007/BF01391200.
↑ 3.0 3.1 Groenewold, H. J. (1946). "On the principles of elementary quantum mechanics". Physica 12 (7): 405–460. doi:10.1016/S0031-8914(46)80059-4.
↑ More directly, note $[x n, p] = i ℏ n x n - 1$ , hence $2 ‖ p ‖ ‖ x ‖ n \geq n ℏ ‖ x ‖ n - 1$ , so that, $\forall n : 2 ‖ p ‖ ‖ x ‖ \geq n ℏ$ . However, $n$ can be arbitrarily large. Utilizing the Weyl relations, below, it can actually be shown that both operators are unbounded.
↑ Townsend, J. S. (2000). A Modern Approach to Quantum Mechanics. Sausalito, CA: University Science Books. ISBN 1-891389-13-0.
↑ 6.0 6.1 Robertson, H. P. (1929). "The Uncertainty Principle". Physical Review 34 (1): 163–164. Bibcode:1929PhRv...34..163R. doi:10.1103/PhysRev.34.163.

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