PX262 - C4b - postulate 4

postulate 4

when a measurement of a dynamical variable represented by the hermitian operator is carried out on a system whose wavefunction is $ψ = \sum_{n} a_{n} ϕ_{n}$ , where $ϕ_{n}$ are the eigenfunctions of $| h a t Q$ corresponding to the eigenvalues $q_{n}$ , then the probability to obtain the result being equal to a particular eigenvalue, $q_{m}$ , will be $| a_{m} |^{2}$

to find the values of the coefficients, $a_{n} :$ $\begin{align*}$ &= \sum\limits_{m}a_{m} \int\phi_{n}^{}\phi_{m},d\tau \
&= \sum\limits_{m}a_{m} \delta_{mn} \
\therefore a_{n} &= \int \phi_{n}^{} \psi ,d\tau
\end{align*}$$
this is called an overlap integral, which is the measurement of the extent to which the two functions are similar