Expectation value (quantum mechanics)

In quantum mechanics, the expectation value is the average result if a system was observed many times.

The expectation value, $\langle {\hat {X}}\rangle$ of an operator, ${\hat {X}}$, operating on a wavefunction $\psi$ is given by

$\langle {\hat {X}}\rangle ={\frac {\int _{-\infty }^{\infty }~\psi ^{*}{\hat {X}}\psi ~dV}{\int _{-\infty }^{\infty }~\psi ^{*}\psi ~dV}}.$

For normalized wavefuntions, $\int _{-\infty }^{\infty }~\psi ^{*}\psi ~dV$ is unity, and the expectation can be expressed in braket notation as

$x=\langle {\hat {X}}\rangle =\langle \psi |{\hat {X}}|\psi \rangle$

See also

• Expected value
• Observable