Operators

Definition and Role

Conceptual Overview

Operator: mathematical entity acting on state vectors in Hilbert space. Role: maps quantum states to quantum states or measurement outcomes. Represents physical quantities (observables).

Mathematical Formulation

Defined as linear transformations: O: H → H, where H is Hilbert space. Operators can be bounded or unbounded. Domain and range subsets of H.

Physical Interpretation

Observable properties correspond to Hermitian operators. Operators encode symmetries, dynamics, and measurement outcomes.

Linear Operators

Linearity Property

Definition: O(α|ψ⟩ + β|φ⟩) = αO|ψ⟩ + βO|φ⟩. Essential for superposition principle. Ensures operator preserves vector space structure.

Bounded vs. Unbounded

Bounded: operator norm finite. Unbounded: common for momentum, position. Requires domain specification. Self-adjointness often involves unbounded operators.

Examples

Identity operator I: leaves state unchanged. Ladder operators in harmonic oscillator: raise/lower energy levels.

Hermitian Operators

Definition

Hermitian (self-adjoint): O = O†. Ensures real eigenvalues. Represents measurable observables.

Properties

Eigenvalues real. Eigenvectors orthogonal. Spectral theorem applies: operator diagonalizable with orthonormal basis.

Examples

Hamiltonian operator (energy), momentum operator, position operator, spin operators.

Operator Algebra

Addition and Scalar Multiplication

Operators form vector space. Sum and scalar multiplication defined pointwise: (O₁ + O₂)|ψ⟩ = O₁|ψ⟩ + O₂|ψ⟩.

Multiplication

Product O₁O₂: composition of operators. Generally non-commutative: O₁O₂ ≠ O₂O₁.

Adjoint Operation

Adjoint O†: defined via inner product: ⟨φ|Oψ⟩ = ⟨O†φ|ψ⟩. Key for defining Hermitian and unitary operators.

Commutators and Compatibility

Definition

Commutator: [A, B] = AB - BA. Measures non-commutativity.

Physical Significance

If [A, B] = 0, observables compatible; simultaneous eigenstates exist. Otherwise, uncertainty relations arise.

Canonical Commutation Relations

Example: [x, p] = iħI. Basis of Heisenberg uncertainty principle.

Eigenvalues and Eigenstates

Eigenvalue Equation

O|ψ⟩ = λ|ψ⟩. λ eigenvalue, |ψ⟩ eigenstate. Physical measurement yields eigenvalues.

Spectral Decomposition

Operator expressed as sum/integral over eigenvalues and projectors: O = ∑ λₙ Pₙ.

Degeneracy

Multiple eigenstates share eigenvalue. Important in symmetry and quantum numbers.

Operators as Observables

Measurement Postulate

Observable corresponds to Hermitian operator. Measurement outcome: eigenvalue λ with probability given by projection.

Expectation Value

⟨O⟩ = ⟨ψ|O|ψ⟩. Statistical mean of measurement results.

Uncertainty

Variance: ⟨(O - ⟨O⟩)²⟩. Linked to commutators and Heisenberg relations.

Unitary Operators and Time Evolution

Definition

Unitary U: U†U = UU† = I. Preserve inner product and norm.

Time Evolution Operator

Evolves states: |ψ(t)⟩ = U(t, t₀)|ψ(t₀)⟩. Generated by Hamiltonian.

Properties

Invertible, norm-preserving, reversible transformations.

Projection Operators

Definition

Projector P: P² = P, P = P†. Projects onto subspace.

Role in Measurement

Measurement collapses state into eigenspace via projectors.

Orthogonality and Completeness

Projectors for distinct eigenvalues are orthogonal. Sum to identity operator.

Operator Representations

Matrix Representation

Operators represented as matrices in chosen basis. Action: matrix multiplication.

Position and Momentum Representations

Operators act as differential operators in position/momentum basis.

Dirac Notation

Abstract representation with bras and kets. Operator expressed as sum of outer products.

Measurement Postulate

Collapse of the Wavefunction

Measurement projects state onto eigenstate of measured operator.

Probability Rule

Outcome probability: p(λ) = |⟨ψ|φ_λ⟩|², φ_λ eigenstate.

Post-measurement State

State immediately after measurement is eigenstate associated to outcome.

Applications in Quantum Mechanics

Quantum Harmonic Oscillator

Operators: ladder, Hamiltonian. Spectrum and eigenstates derived via operators.

Spin Systems

Spin operators generate SU(2) algebra. Measurement of spin components.

Quantum Information

Operators represent quantum gates, observables, and measurement procedures.

Time Evolution Operator:U(t, t₀) = exp(-iH(t - t₀)/ħ)Where:H = Hamiltonian operatorħ = reduced Planck constantt, t₀ = times

Commutation Relation Example:[x, p] = xp - px = iħIWhere:x = position operatorp = momentum operatorI = identity operatorħ = reduced Planck constant

References

• J. J. Sakurai, J. Napolitano, Modern Quantum Mechanics, 2nd ed., Addison-Wesley, 2011, pp. 45-110.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958, pp. 60-85.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, pp. 200-250.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998, pp. 150-200.
• M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000, pp. 100-140.