Expectation Value of Dynamic Variables | Quantum Mechanics

It is defined as the average of the result of a large number of independent measurements of a physical observable on the same system.

$\prec\hat{A}\succ=\frac{\intop_{-\infty}^{\infty}\psi^{*}\hat{A}\psi dx}{\intop_{-\infty}^{\infty}\psi^{*}\psi dx}=\frac{\prec\psi|\hat{A}|\phi\succ}{\prec\psi|\psi\succ}$

here, ${\prec\psi|\psi\succ}$ =  $\left(Norm\right)^{2}$ = ||$\psi$|| = 1(if $\psi$ $\rightarrow$ normalized)

Note:

(1) If the state of the particle is an eigenfunction of the operator Â, then the expectation value of the physical observable corresponding to  will be equal to the eigenvalue of  corresponding to the state of the particle.

(2) The state of the particle is given as:   

  $|\psi\succ=C_{1}|\phi_{1}\succ+C_{2}|\phi_{2}\succ+C_{3}|\phi_{3}\succ+……=$

$\sum_{n}C_{n}|\phi_{n}\succ$

where

$|\phi_{1}\succ,|\phi_{2}\succ…….$

are the eigenfunction of the operator Â, then expectation value of the physical observable corresponding to  will be

$\prec\hat{A}\succ=\sum_{n}|C_{n}|^{2}\lambda_{n}$

where $\lambda_{n}$ are the eigenvalues of operator $Â$ corresponding to

$|\phi_{n}\succ$