Physlib

Physlib.QuantumMechanics.Operators.Commutation

Commutation relations

i. Overview

In this module we compute the commutators for common operators acting on Schwartz maps on `Space d`.

Commutator lemmas come in three flavors: - 1. `a_commutation_b` lemmas are of the form `⁅a, b⁆ = (⋯)`. - 2. `a_comp_b_commute` and `a_comp_commute` lemmas are of the form `a ∘ b = b ∘ a`. - 3. `a_comp_b_eq` lemmas are of the form `a ∘ b = b ∘ a + (⋯)`.

ii. Key results

- `position_commutation_momentum` : The canonical commutation relations. - `angularMomentum_commutation_position` : The position operator transforms as a vector under infinitessimal rotations. - `angularMomentum_commutation_radiusRegPow` : Functions of `‖x‖²` commute with the angular momenta. - `angularMomentum_commutation_momentum` : The momentum operator transforms as a vector under infinitessimal rotations. - `angularMomentum_commutation_angularMomentum` : Angular momenta generate an `𝔰𝔬(d)` algebra. - `angularMomentumSqr_commutation_angularMomentum` : `𝐋²` is a quadratic Casimir of `𝔰𝔬(d)`.

iii. Table of contents

- A. General - B. Commutators - B.1. Position / position - B.2. Momentum / momentum - B.3. Position / momentum - B.4. Angular momentum / position - B.5. Angular momentum / momentum - B.6. Angular momentum / angular momentum

iv. References

A. General

B. Commutators

B.1. Position / position

B.2. Momentum / momentum

B.3. Position / momentum

B.4. Angular momentum / position

B.5. Angular momentum / momentum

B.6. Angular momentum / angular momentum

2 declarations

theorem

The identity operator II commutes with any operator AA, i.e., [I,A]=0[I, A] = 0

Let S(Rd,C)\mathcal{S}(\mathbb{R}^d, \mathbb{C}) be the Schwartz space of complex-valued functions on the dd-dimensional Euclidean space Rd\mathbb{R}^d. For any continuous linear operator A:S(Rd,C)→S(Rd,C)A: \mathcal{S}(\mathbb{R}^d, \mathbb{C}) \to \mathcal{S}(\mathbb{R}^d, \mathbb{C}), the commutator of the identity operator II and AA is zero, i.e., [I,A]=0 [I, A] = 0 where [I,A]=I∘A−A∘I[I, A] = I \circ A - A \circ I.

theorem

Pointwise Canonical Commutation Relation xi(p^iψ)(x)−p^i(xiψ)(x)=iℏψ(x)x_i (\hat{p}_i \psi)(x) - \hat{p}_i (x_i \psi)(x) = i\hbar \psi(x)

For any Schwartz function ψ∈S(Rd,C)\psi \in \mathcal{S}(\mathbb{R}^d, \mathbb{C}) and any point x∈Rdx \in \mathbb{R}^d, the canonical commutation relation between the ii-th position and momentum operators holds pointwise: xi(p^iψ)(x)−(p^i(x^iψ))(x)=iℏψ(x) x_i (\hat{p}_i \psi)(x) - (\hat{p}_i (\hat{x}_i \psi))(x) = i\hbar \psi(x) where xix_i denotes the ii-th coordinate of xx, p^i\hat{p}_i is the ii-th momentum operator, x^i\hat{x}_i is the ii-th position operator, ℏ\hbar is the reduced Planck constant, and ii is the imaginary unit.