Table of Contents
- Introduction
- What Are Operators in Quantum Mechanics?
- Definition of Hermitian Operators
- Mathematical Properties
- Physical Significance
- Hermitian vs Non-Hermitian Operators
- Spectral Theorem and Diagonalization
- Examples of Hermitian Operators
- Eigenvalues and Eigenfunctions
- Inner Product Space and Adjoint Operators
- Self-Adjoint Extensions
- Role in the Measurement Postulate
- Functional Calculus and Operator Functions
- Commutation Relations and Observables
- Hermitian Operators in Quantum Information
- Conclusion
1. Introduction
In quantum mechanics, physical observables — quantities that can be measured — are represented by Hermitian operators. These operators are central to the structure of quantum theory, ensuring that measurable outcomes are real numbers and that systems can be described in terms of well-behaved eigenstates.
2. What Are Operators in Quantum Mechanics?
Operators act on quantum states in Hilbert space and correspond to physical processes or measurements. Common operators include:
- Position: \( \hat{x} \)
- Momentum: \( \hat{p} \)
- Hamiltonian: \( \hat{H} \)
Operators generalize classical functions in the quantum context.
3. Definition of Hermitian Operators
An operator \( \hat{A} \) is Hermitian (or self-adjoint) if:
\[
\langle \phi | \hat{A} \psi \rangle = \langle \hat{A} \phi | \psi \rangle
\]
Or equivalently:
\[
\hat{A}^\dagger = \hat{A}
\]
Where \( \hat{A}^\dagger \) is the adjoint (conjugate transpose in finite dimensions).
4. Mathematical Properties
Hermitian operators satisfy:
- Real eigenvalues
- Orthonormal eigenvectors
- Spectral decomposition (can be diagonalized)
- Complete basis of eigenfunctions in Hilbert space
5. Physical Significance
Only Hermitian operators are associated with physical observables because:
- Measurements yield real numbers
- Measurement outcomes correspond to eigenvalues
- Quantum state collapses to eigenstates upon measurement
6. Hermitian vs Non-Hermitian Operators
| Property | Hermitian | Non-Hermitian |
|---|---|---|
| Adjoint | \( \hat{A}^\dagger = \hat{A} \) | \( \hat{A}^\dagger \ne \hat{A} \) |
| Eigenvalues | Real | Complex (in general) |
| Physical meaning | Observable | Often auxiliary or non-physical |
7. Spectral Theorem and Diagonalization
The spectral theorem states:
Any Hermitian operator \( \hat{A} \) can be written as:
\[
\hat{A} = \sum_n a_n |a_n\rangle \langle a_n|
\]
Where:
- \( a_n \): eigenvalues
- \( |a_n\rangle \): orthonormal eigenvectors
In infinite dimensions, the sum becomes an integral over a continuous spectrum.
8. Examples of Hermitian Operators
- Position operator \( \hat{x} \): acts by multiplication
- Momentum operator \( \hat{p} = -i\hbar \frac{d}{dx} \)
- Hamiltonian \( \hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) \)
- Spin operators: Pauli matrices \( \hat{\sigma}_x, \hat{\sigma}_y, \hat{\sigma}_z \)
All yield real measurement results and satisfy the Hermitian condition.
9. Eigenvalues and Eigenfunctions
Let:
\[
\hat{A} |\psi\rangle = a |\psi\rangle
\]
Then:
- \( a \in \mathbb{R} \)
- \( |\psi\rangle \) is normalized and orthogonal to other eigenstates
- Eigenfunctions form a complete basis
10. Inner Product Space and Adjoint Operators
The adjoint of an operator \( \hat{A} \) is defined via the inner product:
\[
\langle \phi | \hat{A} \psi \rangle = \langle \hat{A}^\dagger \phi | \psi \rangle
\]
In coordinate representation:
- \( \hat{A} = -i\hbar \frac{d}{dx} \): Hermitian on domain of square-integrable functions with suitable boundary conditions
11. Self-Adjoint Extensions
Some differential operators are formally Hermitian but need domain specification to be truly self-adjoint. This is essential in:
- Quantum wells
- Infinite domains
- Quantum field theory
12. Role in the Measurement Postulate
Upon measurement of observable \( \hat{A} \):
- Result is one of the eigenvalues \( a \)
- State collapses to eigenvector \( |a\rangle \)
- Probability of \( a \):
\[
P(a) = |\langle a | \psi \rangle|^2
\]
This process relies on the Hermitian nature of \( \hat{A} \).
13. Functional Calculus and Operator Functions
Functions of Hermitian operators (e.g., \( f(\hat{H}) \)) are defined via spectral decomposition:
\[
f(\hat{A}) = \sum_n f(a_n) |a_n\rangle \langle a_n|
\]
Used in:
- Time evolution: \( e^{-i\hat{H}t/\hbar} \)
- Propagators
- Quantum statistical mechanics
14. Commutation Relations and Observables
Hermitian operators define algebraic structures:
\[
[\hat{x}, \hat{p}] = i\hbar
\]
- Basis of Heisenberg algebra
- Lead to uncertainty principles and canonical quantization
15. Hermitian Operators in Quantum Information
- Qubit observables: Pauli matrices are Hermitian
- Density matrices: Hermitian, positive-semidefinite, trace one
- Quantum gates: generated via exponentials of Hermitian operators
Hermitian matrices define measurements and entanglement criteria.
16. Conclusion
Hermitian operators are indispensable in quantum mechanics. They represent observables, guarantee real outcomes, and provide a basis for understanding measurement, uncertainty, and evolution. Their mathematical properties ensure that quantum theory remains both predictive and internally consistent. A deep grasp of Hermitian operators is essential for mastering quantum systems.
