Definition and Basic Properties
Hermitian Operators Defined
Operator \( A \) on Hilbert space \( \mathcal{H} \) is Hermitian (self-adjoint) if \( A = A^\dagger \), where \( A^\dagger \) is the adjoint. Domain \( D(A) = D(A^\dagger) \). Ensures real expectation values.
Linearity and Domain
Hermitian operators are linear: \( A(\alpha \psi + \beta \phi) = \alpha A\psi + \beta A\phi \). Domains are dense subsets of \( \mathcal{H} \) to guarantee well-defined adjoints.
Bounded vs Unbounded
Hermitian operators may be bounded (norm finite) or unbounded (e.g. momentum operator). Unbounded operators require careful domain specification.
Mathematical Structure
Adjoint Operator
Adjoint \( A^\dagger \) defined by \( \langle A\psi | \phi \rangle = \langle \psi | A^\dagger \phi \rangle \) for all \( \psi, \phi \in \mathcal{H} \). Hermiticity demands \( A = A^\dagger \).
Domain Considerations
Operator domains must satisfy \( D(A) = D(A^\dagger) \). Essential for self-adjointness rather than mere symmetry.
Closedness and Self-Adjointness
Hermitian operators are closed or closable. Self-adjointness implies closedness; crucial for spectral analysis.
Physical Interpretation
Observables in Quantum Mechanics
Physical observables correspond to Hermitian operators. Eigenvalues represent possible measurement outcomes.
Expectation Values
Expectation value \( \langle \psi | A | \psi \rangle \) is real for Hermitian \( A \), consistent with physical quantities.
Probability and Measurement
Eigenstates form measurement bases. Probability of outcome linked to projection of state onto eigenvector.
Eigenvalues and Eigenvectors
Real Eigenvalues
Hermitian operators have purely real eigenvalues: \( A|\phi\rangle = \lambda |\phi\rangle, \lambda \in \mathbb{R} \).
Orthogonality
Eigenvectors corresponding to distinct eigenvalues are orthogonal: \( \langle \phi_i | \phi_j \rangle = 0, i \neq j \).
Completeness
Eigenvectors form a complete basis in \( \mathcal{H} \), enabling spectral decomposition.
Spectral Theorem
Statement
Every Hermitian operator admits a spectral decomposition: \( A = \int \lambda dE(\lambda) \), where \( E(\lambda) \) is a projection-valued measure.
Projection-Valued Measures
Families of orthogonal projections satisfying \( E(\lambda)E(\mu) = E(\min(\lambda,\mu)) \), crucial for representing observables.
Functional Calculus
Enables defining functions \( f(A) = \int f(\lambda) dE(\lambda) \), extending operator functions beyond polynomials.
A = ∫ λ dE(λ)f(A) = ∫ f(λ) dE(λ) Examples of Hermitian Operators
Position Operator \(\hat{x}\)
Defined on \( L^2(\mathbb{R}) \), acts as multiplication: \( (\hat{x}\psi)(x) = x\psi(x) \). Self-adjoint with continuous spectrum \( \mathbb{R} \).
Momentum Operator \(\hat{p}\)
Defined as \( \hat{p} = -i\hbar \frac{d}{dx} \) on suitable domains. Hermitian and self-adjoint with continuous spectrum.
Spin Operators
Finite-dimensional Hermitian matrices acting on spin space. Eigenvalues correspond to spin projections.
| Operator | Definition | Spectrum |
|---|---|---|
| Position \(\hat{x}\) | Multiplication by \(x\) | Continuous, \(\mathbb{R}\) |
| Momentum \(\hat{p}\) | \(-i\hbar \frac{d}{dx}\) | Continuous, \(\mathbb{R}\) |
| Spin \(S_z\) | Pauli matrix \(\sigma_z\) | Discrete, \(\pm \hbar/2\) |
Hermitian vs Symmetric Operators
Symmetric Operators
Operator \(A\) is symmetric if \( \langle A\psi|\phi \rangle = \langle \psi|A\phi \rangle \) for all \( \psi, \phi \in D(A) \), but domain may differ from adjoint's.
Self-Adjointness as Stronger Condition
Hermiticity (self-adjointness) requires \(A = A^\dagger\) including domain equality. Ensures spectral theorem applicability.
Example: Momentum Operator Domains
Momentum operator symmetric on smooth compact support functions but self-adjoint only on specific Sobolev spaces.
Symmetric: A ⊆ A†Self-adjoint: A = A† and D(A) = D(A†) Measurement Postulate and Hermitian Operators
Measurement Outcomes
Possible results: eigenvalues of Hermitian operator representing observable.
State Collapse
Post-measurement state projects onto eigenvector associated with measured eigenvalue.
Probability Rule
Probability of outcome \(\lambda\): \( |\langle \phi_\lambda | \psi \rangle|^2 \), where \( \phi_\lambda \) is eigenvector.
Commutation Relations and Compatibility
Commutators
Operators \(A, B\) commute if \( [A,B] = AB - BA = 0 \). Compatible observables measured simultaneously.
Uncertainty Principle
Non-commuting Hermitian operators imply uncertainty relations: \( \Delta A \Delta B \geq \frac{1}{2}|\langle [A,B] \rangle| \).
Examples
Position and momentum operators satisfy \( [\hat{x}, \hat{p}] = i \hbar \), non-commuting.
Functional Calculus for Hermitian Operators
Definition
Using spectral theorem, define \( f(A) = \int f(\lambda) dE(\lambda) \) for measurable \( f \).
Applications
Exponentials \( e^{iA} \), projectors, and resolvents derived via functional calculus.
Example: Time Evolution
Time propagator \( U(t) = e^{-iHt/\hbar} \) from Hamiltonian \(H\) Hermitian operator.
Extensions and Generalizations
Unbounded Operators
Many physical observables are unbounded Hermitian operators requiring rigged Hilbert space treatment.
Positive Operators
Hermitian operators with non-negative spectrum used in density matrices and quantum states.
Non-Hermitian Generalizations
PT-symmetric and pseudo-Hermitian operators studied beyond standard quantum mechanics framework.
Applications in Quantum Mechanics
Quantum Measurement
Hermitian operators encode measurement outcomes and probabilities in quantum experiments.
Quantum Dynamics
Hamiltonian Hermitian operator governs system evolution via Schrödinger equation.
Quantum Computing
Hermitian operators represent observables and measurement operators in qubit manipulation.
| Application Area | Role of Hermitian Operators |
|---|---|
| Measurement | Define observable outcomes and probabilities |
| Dynamics | Hamiltonian operator generating time evolution |
| Quantum Computing | Observable measurement and state manipulation |
References
- J.J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, 1994, pp. 45-78.
- M. Reed, B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1980, pp. 234-290.
- R. Shankar, Principles of Quantum Mechanics, Springer, 1994, pp. 120-165.
- J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955, pp. 210-260.
- G. Teschl, Mathematical Methods in Quantum Mechanics, AMS, 2009, pp. 150-200.