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 $\begin{pmatrix} 1 \\ 0\end{pmatrix}$

the Hamiltonian can be written as