# Expectation Value of Dynamic Variables | Quantum Mechanics

Quantum Mechanics
Basic Quantum Mechanics

Expectation Value of Dynamic Variables

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 Â will be equal to the eigenvalue of Â 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 Â will be

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

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

$|\phi_{n}\succ$