Question

The normalized wave functions \(\psi_1\) and \(\psi_2\) correspond to the ground state and the first excited state of a particle in a potential. The operator \(\hat{A}\) acts on the wave functions as \(\hat{A}\psi_1 = \psi_2\) and \(\hat{A}\psi_2 = \psi_1\). The expectation value of the operator \(\hat{A}\) for the state \(\psi = (3\psi_1 + 4\psi_2)/5\) is:

• A. 0.96
• B. −0.32
• C. 0.75
• D. 0

Hint:

Apply \(\hat{A}\) (which swaps the states) to \(\psi\), then take the overlap with \(\psi\); the cross terms give \(2\cdot 3\cdot 4/25\).

Answer 1

The Correct Option is A

Step 1: The expectation value is \(\langle A\rangle = \langle\psi|\hat{A}|\psi\rangle\), with \(\psi_1,\psi_2\) orthonormal, so \(\langle\psi_i|\psi_j\rangle=\delta_{ij}\).

Step 2: Act \(\hat{A}\) on the state:
\[\hat{A}\psi = \tfrac{1}{5}\big(3\hat{A}\psi_1 + 4\hat{A}\psi_2\big) = \tfrac{1}{5}\big(3\psi_2 + 4\psi_1\big).\]
Step 3: Form the inner product with \(\psi=\tfrac{1}{5}(3\psi_1+4\psi_2)\):
\[\langle A\rangle = \tfrac{1}{25}\big(3\psi_1+4\psi_2\,\big|\,4\psi_1+3\psi_2\big).\]
Step 4: Only matching terms survive:
\[\langle A\rangle = \tfrac{1}{25}\big(3\cdot 4 + 4\cdot 3\big) = \tfrac{24}{25}.\]
Step 5: Evaluate the fraction.
\[\boxed{\langle A\rangle = 0.96}\]