Expectation value (quantum physics)

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A statistical expectation value is defined as the sum of the values of possible outcomes, multiplied by the probability of that outcome. In quantum mechanics, quantities such as position and momentum are described statistically, so it makes sense to talk about the expected value of the position of a particle, say. Suppose \hat Q is an operator with eigenstates | q \rangle. The expectation value of \hat Q for a normalized state Ψ is given by:

\sum_{q} q \left| \left \langle q | \Psi \right \rangle \right|^2

where the sum becomes an integral if the eigenvalues of \hat Q are continuous. In this formula, q is the value of the observable for a given eigenstate |q\rangle, while the second term is the squared probability amplitude of that eigenstate, i.e., the probability of observing the given value.

This formula can easily be simplified to the form used most frequently in practice:

\sum_{q} q \left| \left \langle q | \Psi \right \rangle \right|^2 = \sum_{q} \left \langle \Psi | \hat Q | q \right \rangle \left \langle q | \Psi \right \rangle = \left \langle \Psi | \hat Q | \Psi \right \rangle = \left \langle \hat Q \right \rangle_{\Psi}

Expectation values are central to the statement of Heisenberg's uncertainty principle, and are used in many other quantum-mechanical theorems. They are particularly useful for deriving analogues of classical physics equations: see for example the quantum virial theorem, or quantum harmonic oscillator coherent states.

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