Question 1
1. About the "particle in a box" problem, consider a state with the equal-weight superposition of two
lowest-energy stationary states, find the probability density function $P(x,t)$ for this state with detailed
calculations.
$t=0 \quad \psi(x,0) = \sqrt{\frac{2}{L}}sin\frac{n\pi x}{L}$
$E_n = \frac{\hbar^2 n^2}{2mL^2}$
Question 2
2. Begin with the time-dependent Schrodinger equation, derive the (time-independent)
stationary-state Schrodinger equation step by step, and discuss the features of the (time-dependent)
stationary states and general states
$i\hbar\frac{\partial \psi(x,t)}{\partial t} = [-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x)]\psi(x,t)$
$\psi(x,t) = F(t)\psi(x)$
