Hermitian Operators: Foundations of Quantum Observables

Table of Contents

1. Introduction
2. What Are Operators in Quantum Mechanics?
3. Definition of Hermitian Operators
4. Mathematical Properties
5. Physical Significance
6. Hermitian vs Non-Hermitian Operators
7. Spectral Theorem and Diagonalization
8. Examples of Hermitian Operators
9. Eigenvalues and Eigenfunctions
10. Inner Product Space and Adjoint Operators
11. Self-Adjoint Extensions
12. Role in the Measurement Postulate
13. Functional Calculus and Operator Functions
14. Commutation Relations and Observables
15. Hermitian Operators in Quantum Information
16. 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.

---