In quantum mechanics, the expectation value is a fundamental concept that represents the average value of an observable quantity for a given quantum state. It provides a way to predict the outcome of measurements in quantum systems.
Mathematically, the expectation value of an observable A is denoted as ⟨A⟩ and is calculated using the wave function (or state vector) of the system. For a discrete observable with eigenvalues a₁, a₂, ..., the expectation value is given by:
⟨A⟩ = ∑ aₙ P(aₙ),
where P(aₙ) is the probability of obtaining the eigenvalue aₙ when measuring the observable A. The probability is obtained from the Born rule, which states that the probability of measuring a particular eigenvalue is proportional to the squared magnitude of the corresponding coefficient of the state vector.
For a continuous observable, the expectation value is calculated using an integral instead of a sum. If the observable has a continuous spectrum with probability density function P(a), the expectation value is given by:
⟨A⟩ = ∫ a P(a) da.
The expectation value provides a prediction of the average outcome of a measurement if the measurement is repeated many times on identically prepared quantum systems. It is an essential concept in quantum mechanics and plays a crucial role in understanding the behavior of quantum systems and making predictions about their properties.