Table of Contents

1. Introduction
2. Operators and Observables
3. Definition of Commutator
4. Mathematical Properties of Commutators
5. Canonical Commutation Relations
6. Position-Momentum Commutation
7. Angular Momentum Commutation Relations
8. Spin and Pauli Matrices
9. Consequences for Measurement
10. Commutators and Uncertainty Principle
11. Lie Algebras and Structure Constants
12. Commutator Algebra in Matrix Mechanics
13. Time Evolution and Commutators
14. Symmetries and Conserved Quantities
15. Applications in Quantum Field Theory
16. Conclusion

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1. Introduction

Commutation relations are fundamental to the algebraic structure of quantum mechanics. They encode the non-commutative nature of quantum observables and lie at the heart of quantum dynamics, measurement, and uncertainty.

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2. Operators and Observables

In quantum mechanics:

• Observables are represented by operators on a Hilbert space
• The order in which operators act matters:
  \[
  \hat{A}\hat{B} \neq \hat{B}\hat{A}
  \]
  This is a key departure from classical physics.

---

3. Definition of Commutator

Given two operators \( \hat{A} \) and \( \hat{B} \), the commutator is defined as:

\[
[\hat{A}, \hat{B}] = \hat{A}\hat{B} – \hat{B}\hat{A}
\]

• If \( [\hat{A}, \hat{B}] = 0 \): operators commute
• If not: they are non-commuting, and their measurements are incompatible

---

4. Mathematical Properties of Commutators

• Linearity:
  \[
  [\hat{A} + \hat{B}, \hat{C}] = [\hat{A}, \hat{C}] + [\hat{B}, \hat{C}]
  \]
• Jacobi Identity:
  \[
  [\hat{A}, [\hat{B}, \hat{C}]] + [\hat{B}, [\hat{C}, \hat{A}]] + [\hat{C}, [\hat{A}, \hat{B}]] = 0
  \]
• Commutator with product:
  \[
  [\hat{A}, \hat{B}\hat{C}] = [\hat{A}, \hat{B}]\hat{C} + \hat{B}[\hat{A}, \hat{C}]
  \]

---

5. Canonical Commutation Relations

The most fundamental commutation relation in quantum mechanics:

\[
[\hat{x}, \hat{p}] = i\hbar
\]

Where:

• \( \hat{x} \): position operator
• \( \hat{p} \): momentum operator

It expresses the non-commutativity of phase space coordinates.

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6. Position-Momentum Commutation

In one dimension:

\[
\hat{x} \psi(x) = x \psi(x), \quad \hat{p} \psi(x) = -i\hbar \frac{d}{dx} \psi(x)
\]

Then:

\[
[\hat{x}, \hat{p}] \psi(x) = i\hbar \psi(x)
\]

Confirms the canonical commutator.

---

7. Angular Momentum Commutation Relations

Angular momentum components \( \hat{L}_x, \hat{L}_y, \hat{L}_z \) obey:

\[
[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z
\]
\[
[\hat{L}_y, \hat{L}_z] = i\hbar \hat{L}_x
\]
\[
[\hat{L}_z, \hat{L}_x] = i\hbar \hat{L}_y
\]

These form the Lie algebra of the rotation group SO(3).

---

8. Spin and Pauli Matrices

Spin-1/2 observables are represented by Pauli matrices:

\[
[\sigma_x, \sigma_y] = 2i\sigma_z, \quad \text{and cyclic permutations}
\]

This reflects the algebra of angular momentum at the quantum level.

---

9. Consequences for Measurement

Non-commuting operators:

• Cannot be simultaneously diagonalized
• Cannot have well-defined values in the same quantum state
• Measurement of one affects the uncertainty of the other

---

10. Commutators and Uncertainty Principle

From:

\[
[\hat{A}, \hat{B}] = i\hat{C}
\]

We get:

\[
\Delta A \Delta B \ge \frac{1}{2} |\langle [\hat{A}, \hat{B}] \rangle|
\]

For position and momentum:

\[
\Delta x \Delta p \ge \frac{\hbar}{2}
\]

---

11. Lie Algebras and Structure Constants

If:

\[
[\hat{T}_a, \hat{T}_b] = i f^{abc} \hat{T}_c
\]

Then \( f^{abc} \) are the structure constants of a Lie algebra. Commutators encode the symmetry algebra of quantum systems.

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12. Commutator Algebra in Matrix Mechanics

Heisenberg’s original formulation was built on:

• Matrix observables
• Operator algebra
• Time evolution governed by commutators:
  \[
  \frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}] + \left(\frac{\partial \hat{A}}{\partial t}\right)
  \]

This is equivalent to Schrödinger’s picture.

---

13. Time Evolution and Commutators

Heisenberg equation of motion:

\[
\frac{d\hat{A}}{dt} = \frac{i}{\hbar}[\hat{H}, \hat{A}]
\]

If \( [\hat{H}, \hat{A}] = 0 \), then \( \hat{A} \) is conserved — i.e., a constant of motion.

---

14. Symmetries and Conserved Quantities

Symmetry operations correspond to unitary operators:

• If \( \hat{U} \) is a symmetry:
  \[
  \hat{U}^\dagger \hat{H} \hat{U} = \hat{H}
  \]
• If \( [\hat{H}, \hat{G}] = 0 \), then \( \hat{G} \) is conserved

This forms the basis of Noether’s theorem in quantum systems.

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15. Applications in Quantum Field Theory

• Creation and annihilation operators obey commutation (bosons) or anticommutation (fermions) relations
• Gauge symmetries are described via commutator algebra of field operators
• Commutators dictate causal structure in relativistic theories

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16. Conclusion

Commutation relations are the algebraic engine of quantum mechanics. They define how observables interact, reveal the structure of quantum symmetries, and give rise to fundamental limits like the uncertainty principle. Understanding and manipulating commutators is essential for analyzing quantum systems in mechanics, information, and field theory.

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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

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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.

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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.

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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

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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

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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 \( \hat{A} \).

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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

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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

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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.

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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.

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Table of Contents

1. Introduction
2. The Role of Observables in Quantum Theory
3. Mathematical Representation of Observables
4. Hermitian Operators and Their Properties
5. Measurement Postulate
6. Eigenvalues as Measurement Outcomes
7. Projection Postulate and State Collapse
8. Expectation Values and Probabilities
9. Compatible and Incompatible Observables
10. Commutators and the Uncertainty Principle
11. Degeneracy and Simultaneous Measurements
12. Measurement in Discrete vs Continuous Systems
13. Generalized Measurements and POVMs
14. Measurement in Quantum Circuits
15. Interpretations and Measurement Problem
16. Conclusion

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1. Introduction

In quantum mechanics, observables represent physical quantities we can measure, such as position, momentum, or energy. Unlike classical physics, the act of measurement in quantum theory plays a central and non-trivial role: it not only reveals information but also affects the system being measured.

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2. The Role of Observables in Quantum Theory

Quantum mechanics does not assign definite values to observables until they are measured. Instead, it provides probabilities for different outcomes based on the system’s state vector. Observables are the bridge between mathematical states and experimental results.

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3. Mathematical Representation of Observables

Observables are represented by Hermitian operators acting on a Hilbert space:

\[
\hat{A}^\dagger = \hat{A}
\]

Key properties:

• Real eigenvalues (physical measurements must be real)
• Orthogonal eigenstates
• Complete spectral decomposition

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4. Hermitian Operators and Their Properties

Let \( \hat{A} \) be an observable. Then:

• \( \hat{A} \) is self-adjoint: \( \hat{A}^\dagger = \hat{A} \)
• If \( \hat{A}|a\rangle = a|a\rangle \), then \( a \in \mathbb{R} \)
• Eigenstates form a complete orthonormal basis

This ensures measurable outcomes are consistent and interpretable.

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5. Measurement Postulate

Upon measurement of observable \( \hat{A} \) in state \( |\psi\rangle \):

• The result is one of the eigenvalues \( a \)
• The system collapses into the corresponding eigenstate \( |a\rangle \)
• The probability of outcome \( a \) is:

\[
P(a) = |\langle a | \psi \rangle|^2
\]

This is the Born rule.

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6. Eigenvalues as Measurement Outcomes

Only the eigenvalues of \( \hat{A} \) are physically observable. All other values are excluded. The spectrum of the operator determines the possible outcomes.

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7. Projection Postulate and State Collapse

After measuring \( \hat{A} \) and finding result \( a \), the system is projected into:

\[
|\psi\rangle \rightarrow \frac{\hat{P}_a |\psi\rangle}{|\hat{P}_a |\psi\rangle|}
\]

Where \( \hat{P}_a = |a\rangle\langle a| \) is the projector. This process is known as wavefunction collapse.

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8. Expectation Values and Probabilities

The expectation value of \( \hat{A} \) in state \( |\psi\rangle \) is:

\[
\langle \hat{A} \rangle = \langle \psi | \hat{A} | \psi \rangle
\]

It represents the average of many measurements over identically prepared systems.

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9. Compatible and Incompatible Observables

Observables \( \hat{A} \) and \( \hat{B} \) are:

• Compatible if they commute:
  \[
  [\hat{A}, \hat{B}] = 0
  \]
  ⇒ They share a common eigenbasis, and can be measured simultaneously.
• Incompatible if \( [\hat{A}, \hat{B}] \ne 0 \):
  ⇒ Subject to uncertainty relations

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10. Commutators and the Uncertainty Principle

For position and momentum:

\[
[\hat{x}, \hat{p}] = i\hbar
\]

This gives rise to the Heisenberg uncertainty principle:

\[
\Delta x \, \Delta p \ge \frac{\hbar}{2}
\]

Incompatible observables cannot be simultaneously known with arbitrary precision.

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11. Degeneracy and Simultaneous Measurements

• Degenerate eigenvalues: multiple eigenstates share the same eigenvalue
• Measurement of a degenerate observable does not fully specify the post-measurement state
• Additional compatible observables can be used to resolve degeneracy

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12. Measurement in Discrete vs Continuous Systems

Discrete:

• E.g., spin, energy levels
• Finite or countable spectrum

Continuous:

• E.g., position, momentum
• Eigenstates are delta-normalized distributions

\[
\langle x’ | x \rangle = \delta(x – x’)
\]

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13. Generalized Measurements and POVMs

In generalized quantum measurements:

• Use POVMs (Positive Operator-Valued Measures)
• Broader framework than projective measurements
• Useful in quantum information and open quantum systems

Each outcome \( i \) is associated with operator \( E_i \), such that:

\[
\sum_i E_i = \hat{I}, \quad E_i \ge 0
\]

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14. Measurement in Quantum Circuits

• Measured qubits collapse into \( |0\rangle \) or \( |1\rangle \)
• Probability dictated by amplitudes
• Used to extract information from quantum algorithms

Example:
If \( |\psi\rangle = \alpha |0\rangle + \beta |1\rangle \), then:

\[
P(0) = |\alpha|^2, \quad P(1) = |\beta|^2
\]

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15. Interpretations and Measurement Problem

• Copenhagen: collapse is fundamental
• Many-worlds: all outcomes occur in separate branches
• Objective collapse models: propose dynamical collapse mechanisms
• QBism: collapse is Bayesian updating of observer knowledge

Measurement remains the most debated aspect of quantum theory.

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16. Conclusion

Observables and measurements lie at the core of quantum mechanics. They define the interaction between abstract state vectors and empirical outcomes. While measurements provide the link to experiment, they also introduce profound conceptual challenges, from wavefunction collapse to the nature of quantum reality. Mastery of this topic is vital for understanding, designing, and interpreting quantum systems.

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