Commutators - quantum-physics

Definition of Commutators

Operator Commutator

Definition: For two operators A and B, the commutator [A,B] is defined as [A,B] = AB - BA. Measures non-commutativity.

Non-commutativity

Property: If [A,B] ≠ 0, operators do not commute. Implies ordering matters in multiplication.

Notation and Conventions

Bracket notation: square brackets [ , ] denote commutator. Linear in each argument. Scalar multiples factor out.

[A,B] = AB - BA

Mathematical Properties

Antisymmetry

[A,B] = -[B,A]. Reversal of order changes sign.

Jacobi Identity

Fundamental identity: [A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0. Ensures consistent algebraic structure.

Linearity

Linear in each argument: [αA + βB, C] = α[A,C] + β[B,C], α, β scalars.

[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0

Physical Interpretation

Operator Ordering and Measurement

Non-zero commutator implies measurement order affects outcomes. Reflects quantum measurement disturbance.

Relation to Observables

Observables correspond to Hermitian operators. Non-commuting observables cannot be simultaneously measured precisely.

Quantum Compatibility

Commuting observables: compatible, can have simultaneous eigenstates. Non-commuting: incompatible, uncertainty relations.

Commutators and Observables

Hermitian Operators

Observables correspond to Hermitian (self-adjoint) operators. Commutators of Hermitian operators are anti-Hermitian.

Simultaneous Eigenstates

Condition: [A,B] = 0 implies A and B share eigenbasis. Enables simultaneous precise measurement.

Measurement Implications

Non-commuting observables imply intrinsic quantum uncertainty, no common eigenstates.

Uncertainty Principle

Heisenberg Uncertainty Relation

Derived from commutator: ΔA ΔB ≥ |½ ⟨[A,B]⟩|. Quantifies measurement precision limits.

Position-Momentum Example

Canonical commutation: [x, p] = iħ. Leads to Δx Δp ≥ ħ/2.

Generalized Uncertainty

Applies to any pair of non-commuting observables. Sets fundamental quantum limits.

ΔA ΔB ≥ \frac{1}{2} \left| \langle [A,B] \rangle \right|

Examples of Commutators

Position and Momentum

[x, p] = iħ I. Fundamental canonical commutator in quantum mechanics.

Angular Momentum Components

[L_x, L_y] = iħ L_z, cyclic permutations. Reflects rotational symmetry algebra.

Spin Operators

[S_i, S_j] = iħ ε_ijk S_k. Spin angular momentum obeys su(2) algebra.

Operators	Commutator	Physical Significance
x, p	[x,p] = iħ	Canonical conjugates
L_x, L_y	[L_x,L_y] = iħ L_z	Angular momentum components
S_x, S_y	[S_x,S_y] = iħ S_z	Spin operators

Commutators in Lie Algebras

Lie Bracket

Commutator defines Lie bracket in algebra of operators. Satisfies antisymmetry and Jacobi identity.

Structure Constants

In basis {T_i}, [T_i, T_j] = f_{ijk} T_k, where f_{ijk} are structure constants.

Applications in Symmetry

Lie algebras generated by commutators describe symmetry groups in quantum mechanics.

[T_i, T_j] = \sum_k f_{ijk} T_k

Role in Measurement Theory

Compatibility of Observables

Commuting observables correspond to compatible measurements, can be simultaneously diagonalized.

Measurement Disturbance

Non-commutivity induces disturbance: measuring A affects subsequent measurement of B if [A,B] ≠ 0.

Quantum State Collapse

Measurement projects state onto eigenstate of observable; ordering matters for non-commuting operators.

Calculation Techniques

Direct Multiplication

Calculate AB and BA explicitly, subtract. Feasible for finite matrices or known operator forms.

Use of Commutation Relations

Apply known commutators to simplify expressions. Useful in angular momentum and ladder operators.

BCH Formula

Baker-Campbell-Hausdorff formula relates exponentials of operators via nested commutators.

e^{A} e^{B} = e^{A + B + \frac{1}{2}[A,B] + \frac{1}{12}([A,[A,B]] + [B,[B,A]]) + \cdots}

Commutators in Quantum Dynamics

Heisenberg Equation of Motion

Time evolution of operator O: dO/dt = (iħ)^{-1} [H,O] + (∂O/∂t). Connects dynamics and commutators.

Conservation Laws

If [H,O] = 0, O is conserved quantity. Commutators identify constants of motion.

Symmetry Generators

Operators generating symmetries commute with Hamiltonian or satisfy defined commutation relations.

Table of Common Commutators

Operator Pair	Commutator	Context
x, p	[x,p] = iħ	Canonical conjugates
L_x, L_y	[L_x, L_y] = iħ L_z	Angular momentum algebra
S_x, S_y	[S_x, S_y] = iħ S_z	Spin operators
a, a†	[a, a†] = 1	Harmonic oscillator ladder

Advanced Topics

Nested Commutators

Higher order commutators appear in perturbation theory, BCH expansions, Magnus expansions.

Commutator Algebra in Quantum Field Theory

Field operators satisfy commutation or anti-commutation relations encoding statistics and causality.

Deformation Quantization

Commutators correspond to Poisson brackets in classical limit; central to phase space quantization.

References

J.J. Sakurai, "Modern Quantum Mechanics", Addison-Wesley, 1994, pp. 30-45.
L.D. Landau and E.M. Lifshitz, "Quantum Mechanics: Non-Relativistic Theory", 3rd ed., Pergamon Press, 1977, pp. 15-20.
M. Reed and B. Simon, "Methods of Modern Mathematical Physics, Vol. I: Functional Analysis", Academic Press, 1980, pp. 180-190.
P.A.M. Dirac, "The Principles of Quantum Mechanics", 4th ed., Oxford University Press, 1958, pp. 35-50.
S. Weinberg, "The Quantum Theory of Fields, Vol. 1: Foundations", Cambridge University Press, 1995, pp. 75-90.

Introduction

Commutators are central to quantum physics, encapsulating the non-commutative nature of operator observables. They quantify the extent to which the order of operations affects outcomes and underpin fundamental principles such as the uncertainty relations and quantum dynamics.

"The non-commutativity of quantum operators reflects the fundamental limits of measurement and the structure of quantum theory." -- P.A.M. Dirac