Quantum Qual Problems

1.) Given a quantum mechanical particle of mass $M$ confined inside a box of sides $a,b,c$. The particle is allowed to move freely between $0 \lt x \lt a, 0\lt y\lt b$ and $0\lt z\lt c$.

• Use the time-independent Schrodinger equation for this problem to obtain the general form for the eigenfunctions of the particle

• Now apply boundary conditions to obtain the specific eigenfunctions and eigenenergies for this specific problem.

• Assume $a=b=c$ and find the first 6 eigenenergies of the problem in terms of the box side length ($a$), the particle mass ($M$) and standard constants. What are their quantum number? Make a sketch of the eigenvalue spectrum, a table listing these eigenenergies and the quantum numbers of all the states that correspond to them.

Solution: Qal_QuantP1S

2.) A system has two energy eigenstate with eigenvalues $w_1$ and $w_2$. Assume that $w_1 \gt w_2$. Representing the enegy eigenstate by $\begin{pmatrix}1 \\ 0\end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1\end{pmatrix}$. The Hamiltonian can be written as $H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}$

a.) We introduce and interaction $H_1$ whose matrix elements, using the above basis vectors, are $H_1= \begin{pmatrix} 0 & v \\ v & 0\end{pmatrix}$ where v is real. Find the exact values of the energies of the new Hamitonlian, $H=H_0 + H_1$

b.) Assume that $w_1 - w_2 \gt \gt |v|$. Use time independent perturbation theory to compute the first and second order corrections to the energy levels of $H_0$ when $H_1$ is treated as a perturbation.

c.) Let $w=w_1=w_2$ in $H_0$ above. Assuming the system is initially in the eigenstate $\begin{pmatrix}1 \\ 0\end{pmatrix}$ of $H_0$, and the interaction $H_1$ is turned on for a finite time interval $T$. Find the probabitlity that the system will be found in the state $H_0= \begin{pmatrix} w_1 & 0 \\ 0 & w_2\end{pmatrix}$ at the end of that time interval.