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: Express expectation value.
\[ \langle O \rangle = \langle \psi | \hat{O} | \psi \rangle \] Substitute the given \(\psi\): \[ \psi = \frac{1}{\sqrt{2}}(\psi_1 + i\psi_2) \]

Step 2: Apply operator.
\[ \hat{O}|\psi\rangle = \frac{1}{\sqrt{2}}(\hat{O}\psi_1 + i\hat{O}\psi_2) = \frac{1}{\sqrt{2}}\left(\psi_2 + \frac{i}{\sqrt{2}}(\psi_1 + \psi_2)\right) \]

Step 3: Compute inner product.
Using orthonormality: \[ \langle \psi | \hat{O} | \psi \rangle = \frac{1}{2\sqrt{2}}(1 - i(\sqrt{2} + 1)) \]

Step 4: Conclusion.
Hence, the expectation value is \(\dfrac{1}{2\sqrt{2}} (1 - i(\sqrt{2} + 1))\).