Observable

Definition

In quantum mechanics, an observable is a physical quantity that can be measured, represented mathematically by a self-adjoint operator on a Hilbert space. Observables are crucial because they relate the mathematical formalism of quantum theory to experimental results, allowing for the interpretation of physical states. The nature of observables connects directly with concepts such as states, measurements, and the properties of quantum systems, providing a framework for understanding phenomena like normal states, superselection sectors, and quantum spin systems.

5 Must Know Facts For Your Next Test

1. Observables correspond to measurable quantities in a physical system, such as position, momentum, and energy.
2. The act of measurement in quantum mechanics causes a state to collapse to one of the eigenstates of the observable being measured.
3. In the context of normal states, observables help define how these states interact with measurement processes in quantum systems.
4. Superselection sectors relate to the decomposition of observables, where certain quantities cannot be superposed due to physical constraints.
5. Quantum spin systems illustrate how observables can be used to describe the behavior of spin particles and their interactions.

Review Questions

• How do observables relate to normal states in quantum mechanics?
  • Observables are fundamental in defining normal states as they describe measurable quantities associated with these states. A normal state can be thought of as a statistical distribution over different possible outcomes of measurements related to an observable. The interaction between normal states and observables gives rise to the expectation values that provide meaningful interpretations of physical phenomena in quantum mechanics.
• Discuss how observables play a role in the Haag-Kastler axioms and their implications for local quantum field theory.
  • The Haag-Kastler axioms establish a framework for formulating local quantum field theories where observables are central. These axioms dictate how observables behave under the principles of locality and causality. They specify how observables are associated with regions of spacetime and define their algebraic structure. This ensures that measurements conducted in spacelike separated regions do not interfere with each other, maintaining consistency within the theory.
• Evaluate the significance of observables in understanding superselection sectors and how this affects quantum measurements.
  • Observables are pivotal in understanding superselection sectors because they help delineate which physical quantities can be coherently superposed within a quantum system. The presence of superselection rules means that certain observables cannot take on arbitrary values across different sectors, leading to restrictions in the types of measurements that can be made. Evaluating these rules highlights the importance of observables in characterizing distinct physical situations in quantum theory and underscores their role in defining the fundamental limits of what can be measured.

Related terms

Self-adjoint operator:

An operator that is equal to its own adjoint, meaning it has real eigenvalues and orthogonal eigenvectors, crucial for representing observables in quantum mechanics.

Expectation value: The average value of an observable measured over many identical preparations of a quantum system, providing insight into the system's expected behavior.

Hilbert space: A complete vector space equipped with an inner product, where quantum states and observables are mathematically described.