Originally Posted by

**DiscreteW** Hey topsquark,

Huge Quantum exam tomorrow! Few practice probs I didn't understand:

1.) Suppose that an atom in a solid is described by $\displaystyle H = H_0 + \beta x^2$, where $\displaystyle H$ is the hamilotonian, and $\displaystyle H_0 ~\mbox{is the simple harmonic oscillators Hamiltonian}$. Also $\displaystyle \beta x^2$ is needed for the vibrations of an atom (in an object). $\displaystyle \beta$ is a real constant. Determine the $\displaystyle n$-th energy eigenvalue $\displaystyle E_n$ of $\displaystyle H$, assuming $\displaystyle H_0$ 's $\displaystyle n$-th eigenvalue $\displaystyle U_n>$ is also the $\displaystyle n$-th[/tex] energy eigenstate of $\displaystyle H = H_0 + \beta x^2$.

2a.) Using quantum mechanics, describe how you'd solve the system of a hydrogen atom. What are the physical results? Mr F says: This is most certainly found in any quantum mechanics textbook .....

b.) Use separation of variables in $\displaystyle (x,y)$ coordinates in order to solve the time-indep. schrodinger equation for a particle (free) that's restricted to a 2-D square box. The PE (potential) of this system is:

$\displaystyle V(x,y) = 0,~ \mbox{when both x and y are between 0 and a}$.

$\displaystyle ........... = \infty,~ \mbox{elsewhere (thus the particle is not able to leave the box)}$

c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $\displaystyle E_2$. (Note, $\displaystyle E_1$ is the lowest level).