Physics Problems & Solutions: Quantum Mechanics

The state $\psi=\frac{1}{\sqrt 6}\psi_{-1} +\frac{1}{\sqrt 2}\psi_1 +\frac{1}{\sqrt 3}\psi_2$ is linear combination of three orthonormal eigenstates of the operator $\hat{O}$ corresponding to eigenvalues -1,1 and 2. What is the expectation value of $\hat{O}$ for this state?

A. 2/3
B. √(7/6)
C. 1
D. 4/3
E. (√3+2√2-1)/√6

(GR0177 #29)

Solution:

The expectation value:

$\begin{aligned} \langle \hat{O}\rangle &=\langle\psi|\hat{O}|\psi\rangle \\&=O_1\left(\frac{1}{\sqrt{6}}\right )^2\langle \psi_{-1}|\psi_{-1}\rangle +O_2\left(\frac{1}{\sqrt 2}\right )^2\langle\psi_1|\psi_1\rangle +O_3\left( \frac{1}{\sqrt{3}}\right )^2\langle\psi_2|\psi_2\rangle \\&= (-1)\frac{1}{6} +(1)\frac{1}{2}+(2)\frac{1}{3}= 1\end{aligned}$

Answer: C

Physics Problems & Solutions

Quantum Mechanics - Expectation Value

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