Expectation Value (quantum Mechanics) - Formalism in Quantum Mechanics

Formalism in Quantum Mechanics

In quantum theory, an experimental setup is described by the observable to be measured, and the state of the system. The expectation value of in the state is denoted as .

Mathematically, is a self-adjoint operator on a Hilbert space. In the most commonly used case in quantum mechanics, is a pure state, described by a normalized vector in the Hilbert space. The expectation value of in the state is defined as

(1) .

If dynamics is considered, either the vector or the operator is taken to be time-dependent, depending on whether the Schrödinger picture or Heisenberg picture is used. The time-dependence of the expectation value does not depend on this choice, however.

If has a complete set of eigenvectors, with eigenvalues, then (1) can be expressed as

(2) .

This expression is similar to the arithmetic mean, and illustrates the physical meaning of the mathematical formalism: The eigenvalues are the possible outcomes of the experiment, and their corresponding coefficient is the probability that this outcome will occur; it is often called the transition probability.

A particularly simple case arises when is a projection, and thus has only the eigenvalues 0 and 1. This physically corresponds to a "yes-no" type of experiment. In this case, the expectation value is the probability that the experiment results in "1", and it can be computed as

(3) .

In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator in quantum mechanics. This operator does not have eigenvalues, but has a completely continuous spectrum. In this case, the vector can be written as a complex-valued function on the spectrum of (usually the real line). For the expectation value of the position operator, one then has the formula

(4) .

A similar formula holds for the momentum operator, in systems where it has continuous spectrum.

All the above formulae are valid for pure states only. Prominently in thermodynamics, also mixed states are of importance; these are described by a positive trace-class operator, the statistical operator or density matrix. The expectation value then can be obtained as

(5) .