5.5 Commutators and compatible observables

Commutators are key to understanding quantum mechanics. They show how operators interact and whether observables can be measured together. This concept is crucial for grasping the uncertainty principle and the limits of simultaneous measurements.

Compatible observables can be measured precisely at the same time, while incompatible ones can't. This idea shapes our understanding of quantum states and how we can describe them. It's a fundamental part of quantum theory's mathematical framework.

Commutators and Observables

Understanding Commutators

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• Commutator represents the difference between the order of operations for two quantum mechanical operators
• Defined mathematically as $[A, B] = AB - BA$ for operators A and B
• Measures the extent to which two operators fail to commute
• Commutator equals zero when operators commute, indicating they can be measured simultaneously
• Non-zero commutator implies operators do not commute, leading to uncertainty in simultaneous measurements
• Plays a crucial role in determining compatibility of observables and formulating uncertainty relations

Compatible and Incompatible Observables

• Compatible observables have operators that commute with each other
• Commuting operators share a common set of eigenstates
• Simultaneous precise measurements of compatible observables can be performed
• Incompatible observables have operators that do not commute
• Non-commuting operators do not share a complete set of common eigenstates
• Simultaneous precise measurements of incompatible observables are not possible
• Position and momentum operators serve as a classic example of incompatible observables
• Angular momentum components in different directions (Lx, Ly, Lz) also demonstrate incompatibility

Complete Set of Commuting Observables

• Set of observables that mutually commute with each other
• Provides a complete description of a quantum system's state
• Number of observables in the set equals the dimensionality of the system's Hilbert space
• Eigenstates of this set form a basis for the Hilbert space
• Any state of the system can be expressed as a superposition of these eigenstates
• Enables unambiguous specification of quantum states
• Examples include (n, l, m) for hydrogen atom and (px, py, pz) for a free particle in 3D space

Uncertainty Principle

Fundamental Concept and Mathematical Formulation

• Uncertainty principle establishes limits on simultaneous measurement precision of certain pairs of physical observables
• Formulated by Werner Heisenberg in 1927
• Mathematically expressed as $\Delta A \Delta B \geq \frac{1}{2}|\langle [A, B] \rangle|$
• ΔA and ΔB represent standard deviations of observables A and B
• Inequality shows inverse relationship between uncertainties of incompatible observables
• Most famous form relates position and momentum: $\Delta x \Delta p \geq \frac{\hbar}{2}$
• Demonstrates fundamental limitation in nature, not a result of measurement imperfection

Implications and Applications

• Simultaneous eigenstates exist only for compatible observables
• Incompatible observables cannot have simultaneous eigenstates due to non-zero commutator
• Limits ability to prepare quantum systems with definite values for incompatible observables
• Impacts various areas of quantum mechanics (atomic structure, quantum computing)
• Leads to phenomena like zero-point energy in quantum harmonic oscillators
• Explains stability of atoms by preventing electrons from collapsing into the nucleus
• Crucial in understanding quantum tunneling and other quantum effects

Generalized Uncertainty Relations

• Extend beyond position-momentum pair to other observable combinations
• Robertson uncertainty relation generalizes Heisenberg's principle: $\Delta A \Delta B \geq \frac{1}{2}|\langle \psi|[A,B]|\psi \rangle|$
• Applies to any pair of Hermitian operators A and B
• Schrödinger uncertainty relation further refines this: $(\Delta A)^2(\Delta B)^2 \geq \frac{1}{4}|\langle [A,B] \rangle|^2 + \frac{1}{4}|\langle \{A-\langle A \rangle, B-\langle B \rangle\} \rangle|^2$
• Incorporates anticommutator term for more comprehensive uncertainty description
• Useful in analyzing various quantum systems (angular momentum components, energy-time uncertainty)
• Provides deeper insight into the fundamental nature of quantum mechanical measurements