Particle in a 3D box Questions

1. Write (do not derive) the formula for the energy levels of a particle in a three-dimensional cubical box of side $L$. How many electrons can occupy the level having energy $66{\mathrm{\ }h}^2/8{\mathrm{\ }mL}^2$. [Mains 2008]

2. The wave function of a particle confined in a cube of volume $L^3$ is given by $\psi(x,y,z)= \left(\frac{2}{\mathrm{\ }L}\right)^{\frac{3}{2}} \sin{\frac{\pi x}{L}} \sin{\frac{\pi y}{L}} \sin{\frac{\pi z}{L}} .$ Calculate the average values of $p_x$ and $p_x^2$ in the region $0<x<L$.

3. Solve the Schrodinger equation for a particle in a three-dimensional rectangular potential barrier. Explain the terms degenerate and non-degenerate states in this context. [Mains 2015]

4. Determine the discrete energy levels and the corresponding eigenfunctions for a particle in an infinitely deep potential well inside a cube of dimension $L$, assume $V(x,y,z) = \begin{cases} 0 &\text{for } 0<x<L;0< y<L;0<z<L \\ \infty &\text{elsewhere} \end{cases}$ [Mains 2007]