Expectation Value
Introduction
The expectation value is a fundamental concept in quantum mechanics, a branch of physics that describes phenomena on a microscopic scale, such as atoms and subatomic particles. It is a statistical term that provides a measure of the average outcome of a quantum experiment. In other words, if an experiment were to be repeated many times, the expectation value represents the average of all the results.
Definition
In quantum mechanics, the expectation value of an observable quantity, such as position or momentum, is calculated using the state vector of the system and the operator corresponding to the observable. The state vector, represented by the Greek letter psi (Ψ), describes the state of the quantum system, while the operator, typically denoted by a hat (^), is a mathematical function that transforms the state vector.
The expectation value of an observable A for a system in state Ψ is given by the formula:
⟨A⟩ = ⟨Ψ|A^|Ψ⟩
where ⟨Ψ| and |Ψ⟩ are the bra and ket vectors, respectively, in the bra-ket notation introduced by physicist Paul Dirac. The bra vector ⟨Ψ| is the complex conjugate of the ket vector |Ψ⟩.
Calculation
The calculation of expectation values involves the principles of linear algebra and complex numbers. The bra-ket notation simplifies these calculations by providing a concise and intuitive way to represent complex vector spaces.
For a discrete quantum system, the expectation value of an observable A is calculated by summing the product of the probability of each state and the value of A in that state. If the system is in a superposition of states, the expectation value represents the weighted average of the values of A in each state, with the weights being the probabilities of the states.
For a continuous quantum system, the expectation value is calculated by integrating the product of the wave function, the value of A, and the complex conjugate of the wave function, over all space.
Role in Quantum Mechanics
The concept of expectation value plays a crucial role in quantum mechanics. It bridges the gap between the probabilistic nature of quantum mechanics and the deterministic nature of classical physics. While individual measurements in a quantum system can yield any of a range of values, the average of many measurements (the expectation value) behaves in a predictable and deterministic manner, much like quantities in classical physics.
Expectation values are used in various aspects of quantum mechanics, such as the calculation of average properties of quantum systems, the formulation of quantum statistical mechanics, and the development of quantum theories of measurement.