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Concept
Expectation value (quantum mechanics)
Summary
In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring (e.g. measurements which can only yield integer values may have a non-integer mean). It is a fundamental concept in all areas of quantum physics.
Consider an operator . The expectation value is then in Dirac notation with a normalized state vector.
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
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 evolution of the expectation value does not depend on this choice, however.
If has a complete set of eigenvectors , with eigenvalues , then () can be expressed as
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
In quantum theory, it is also possible for an operator to have a non-discrete spectrum, such as the position operator in quantum mechanics.
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