Question

A particle of mass \( m \) is in an infinite square well potential of length \( L \). It is in a superposed state of the first two energy eigenstates, as given by \( \psi(x) = \frac{2}{\sqrt{3L}} \psi_1(x) + \frac{2}{\sqrt{3L}} \psi_2(x) \). Identify the correct statement(s). \( h \) is Planck’s constant.

• A. \( \langle p \rangle = 0 \)
• B. \( \Delta p = \frac{\sqrt{3}h}{2L} \)
• C. \( \langle E \rangle = \frac{3h^2}{8mL^2} \)
• D. \( \Delta x = 0 \)

Hint:

In superposed quantum states, the expectation values of quantities like momentum can be zero, and uncertainties can be found by calculating \( \Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2} \).

Answer 1

The Correct Option is A, B, C

Step 1: Particle in infinite square well potential.
The energy eigenstates in an infinite square well are given by \[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right), \] with corresponding energies \[ E_n = \frac{n^2 \pi^2 h^2}{2mL^2}. \] Step 2: Superposed state.
The superposed state is \[ \psi(x) = \frac{2}{\sqrt{3L}} \left( \psi_1(x) + \psi_2(x) \right). \] Step 3: Expectation value of momentum.
Since the wavefunctions \( \psi_1(x) \) and \( \psi_2(x) \) are odd and even, respectively, the expectation value of momentum \( \langle p \rangle \) for a superposition of odd and even states is zero. Thus, \[ \langle p \rangle = 0. \] Step 4: Uncertainty in momentum.
The uncertainty in momentum \( \Delta p \) can be calculated as \[ \Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2} = \frac{\sqrt{3}h}{2L}. \] Step 5: Expectation value of energy.
The expectation value of energy is \[ \langle E \rangle = \frac{E_1 + E_2}{2} = \frac{3h^2}{8mL^2}. \] Step 6: Final Answer.
Hence, the correct answers are (A) and (B).