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Examples for Expectation Values

For a quantum mechanical system described by a normalized wave function $ \Psi({\bf x},t)$, the expectation value of an operator $ \hat{O}({\bf\hat{x}},{\bf\hat{p}})$ is

$\displaystyle \langle O({\bf\hat{x}, \hat{p}}) \rangle_t$ $\displaystyle =$ $\displaystyle \int d^dx \Psi^*({\bf x},t) O({\bf\hat{x}, \hat{p}}) \Psi({\bf x},t).$ (182)

Here, $ d$ denotes the dimension of the configuration space. For example, in $ d=1$ dimension we have
$\displaystyle \langle x \rangle_t$ $\displaystyle =$ $\displaystyle \int dx \Psi^*(x,t) x \Psi(x,t),$   position  
$\displaystyle \langle p \rangle_t$ $\displaystyle =$ $\displaystyle \int dx \Psi^*(x,t) \frac{\hbar}{i} \frac{\partial}{\partial x} \Psi(x,t),$   position (183)


Tobias Brandes 2004-02-04