Question

A linear operator \(\hat{O}\) acts on two orthonormal states of a system \(\psi_1\) and \(\psi_2\) as per the following: \[ \hat{O}\psi_1 = \psi_2, \hat{O}\psi_2 = \frac{1}{\sqrt{2}}(\psi_1 + \psi_2) \] The system is in a superposed state defined by \[ \psi = \frac{1}{\sqrt{2}}\psi_1 + \frac{i}{\sqrt{2}}\psi_2 \] The expectation value of \(\hat{O}\) in the state \(\psi\) is:

• A. \(\dfrac{1}{2\sqrt{2}} (1 + i(\sqrt{2} + 1))\)
• B. \(\dfrac{1}{2\sqrt{2}} (1 - i(\sqrt{2} + 1))\)
• C. \(\dfrac{1}{2\sqrt{2}} (1 + i(\sqrt{2} - 1))\)
• D. \(\dfrac{1}{2\sqrt{2}} (1 - i(\sqrt{2} - 1))\)

Hint:

When finding expectation values, always use orthonormality (\(\langle \psi_i | \psi_j \rangle = \delta_{ij}\)) to simplify calculations.

Answer 1

The Correct Option is D

Step 1: Calculate $\hat{O}|\psi\rangle$

$$\hat{O}|\psi\rangle = \hat{O}\left(\frac{1}{\sqrt{2}}\psi_1 + \frac{i}{\sqrt{2}}\psi_2\right)$$

$$= \frac{1}{\sqrt{2}}\hat{O}\psi_1 + \frac{i}{\sqrt{2}}\hat{O}\psi_2$$

$$= \frac{1}{\sqrt{2}}\psi_2 + \frac{i}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}(\psi_1 + \psi_2)$$

$$= \frac{1}{\sqrt{2}}\psi_2 + \frac{i}{2}(\psi_1 + \psi_2)$$

$$= \frac{i}{2}\psi_1 + \left(\frac{1}{\sqrt{2}} + \frac{i}{2}\right)\psi_2$$

Step 2: Calculate $\langle\psi|$

$$\langle\psi| = \frac{1}{\sqrt{2}}\langle\psi_1| + \frac{-i}{\sqrt{2}}\langle\psi_2|$$

(Note: complex conjugate of $i$ is $-i$)

Step 3: Calculate $\langle\psi|\hat{O}|\psi\rangle$

$$\langle\psi|\hat{O}|\psi\rangle = \left(\frac{1}{\sqrt{2}}\langle\psi_1| + \frac{-i}{\sqrt{2}}\langle\psi_2|\right)\left[\frac{i}{2}\psi_1 + \left(\frac{1}{\sqrt{2}} + \frac{i}{2}\right)\psi_2\right]$$

Using orthonormality $\langle\psi_i|\psi_j\rangle = \delta_{ij}$:

$$= \frac{1}{\sqrt{2}} \cdot \frac{i}{2}\langle\psi_1|\psi_1\rangle + \frac{1}{\sqrt{2}}\left(\frac{1}{\sqrt{2}} + \frac{i}{2}\right)\langle\psi_1|\psi_2\rangle$$

$$+ \frac{-i}{\sqrt{2}} \cdot \frac{i}{2}\langle\psi_2|\psi_1\rangle + \frac{-i}{\sqrt{2}}\left(\frac{1}{\sqrt{2}} + \frac{i}{2}\right)\langle\psi_2|\psi_2\rangle$$

$$= \frac{i}{2\sqrt{2}} + 0 + 0 + \frac{-i}{\sqrt{2}}\left(\frac{1}{\sqrt{2}} + \frac{i}{2}\right)$$

$$= \frac{i}{2\sqrt{2}} + \frac{-i}{2} + \frac{-i^2}{2\sqrt{2}}$$

$$= \frac{i}{2\sqrt{2}} - \frac{i}{2} + \frac{1}{2\sqrt{2}}$$

$$= \frac{1}{2\sqrt{2}} + \frac{i}{2\sqrt{2}} - \frac{i}{2}$$

$$= \frac{1}{2\sqrt{2}} + i\left(\frac{1}{2\sqrt{2}} - \frac{1}{2}\right)$$

$$= \frac{1}{2\sqrt{2}}\left(1 + i\left(1 - \sqrt{2}\right)\right)$$

$$= \frac{1}{2\sqrt{2}}\left(1 - i(\sqrt{2} - 1)\right)$$

Answer: (D) $\frac{1}{2\sqrt{2}}(1 - i(\sqrt{2} - 1))$