Problem of the Week

A particle of mass m moves in a one-dimensional infinite square well of width L:

$$V(x) = \begin{cases} 0 & 0 \leq x \leq L \ \infty & \text{otherwise} \end{cases}$$

Part A: Write down the normalised energy eigenfunctions \(\psi_n(x)\) and the corresponding eigenvalues \(E_n\).

Part B: If the particle is initially prepared in the state \(\psi(x,0) = A x(L-x)\), find the normalisation constant \(A\) and express this state as a superposition of energy eigenfunctions.

Part C: Calculate the time evolution \(\psi(x,t)\) and find the probability that a measurement of the energy at time \(t\) yields the ground state \(E_1\).