Physlib

Physlib.QuantumMechanics.Operators.AngularMomentum

Angular momentum operator

i. Overview

In this module we introduce several angular momentum operators for quantum mechanics on `Space d`.

ii. Key results

Definitions: - `angularMomentumOperator` : (components of) the angular momentum operator acting on Schwartz maps `𝓢(Space d, ℂ)` as `𝐱ᵢ∘𝐩ⱼ - 𝐱ⱼ∘𝐩ᵢ`. - `angularMomentumOperatorSqr` : the operator acting on Schwartz maps `𝓢(Space d, ℂ)` as `½ ∑ᵢⱼ 𝐋ᵢⱼ∘𝐋ᵢⱼ`. - `angularMomentumOperator2D` : the (pseudo)scalar angular momentum operator for `d = 2`. - `angularMomentumOperator3D` : the (pseudo)vector angular momentum operator for `d = 3`.

Notation: - `𝐋` for `angularMomentumOperator` - `𝐋²` for `angularMomentumOperatorSqr`

iii. Table of contents

- A. Angular momentum operator - A.1 Antisymmetry - B. Angular momentum squared operator - C. Special cases in low dimensions

iv. References

A. Angular momentum operator

A.1 Antisymmetry

B. Angular momentum squared operator

C. Special cases in low dimensions

• d = 1 : The angular momentum operator is trivial.

• d = 2 : The angular momentum operator has only one independent component, 𝐋₀₁, which may be thought of as a (pseudo)scalar operator.

• d = 3 : The angular momentum operator has three independent components, 𝐋₀₁, 𝐋₁₂ and 𝐋₂₀. Dualizing using the Levi-Civita symbol produces the familiar (pseudo)vector angular momentum operator with components 𝐋₀ = 𝐋₁₂, 𝐋₁ = 𝐋₂₀ and 𝐋₂ = 𝐋₀₁.

3 declarations

definition

Notation for the angular momentum operator Lij\mathbf{L}_{ij}

The definition `QuantumMechanics.term𝐋` provides the notation for the angular momentum operator components Lij\mathbf{L}_{ij}. In quantum mechanics, these components are defined as Lij=xipj−xjpi\mathbf{L}_{ij} = \mathbf{x}_i \mathbf{p}_j - \mathbf{x}_j \mathbf{p}_i, where xi\mathbf{x}_i is the ii-th position operator and pj\mathbf{p}_j is the jj-th momentum operator acting on the space of Schwartz functions S(Rd,C)\mathcal{S}(\mathbb{R}^d, \mathbb{C}).

definition

Notation for angular momentum operator components L[i,j]\mathbf{L}[i, j]

This definition introduces the notation L[i,j]\mathbf{L}[i, j] for the (i,j)(i, j)-th component of the angular momentum operator. It corresponds to the operator defined by L^ij=xip^j−xjp^i\hat{L}_{ij} = x_i \hat{p}_j - x_j \hat{p}_i acting on the space of Schwartz functions S(Rd,C)\mathcal{S}(\mathbb{R}^d, \mathbb{C}), where xix_i is the position operator and p^j\hat{p}_j is the momentum operator.

definition

Notation for L2\mathbf{L}^2

The notation L2\mathbf{L}^2 represents the squared angular momentum operator. As defined in the context of quantum mechanics on a dd-dimensional space, it acts on the space of Schwartz functions S(Rd,C)\mathcal{S}(\mathbb{R}^d, \mathbb{C}) according to the formula L2=12∑i,jLij∘Lij\mathbf{L}^2 = \frac{1}{2} \sum_{i,j} \mathbf{L}_{ij} \circ \mathbf{L}_{ij}, where Lij\mathbf{L}_{ij} are the components of the angular momentum operator.