Quantum Qual Problems

1.) Given a quantum mechanical particle of mass confined inside a box of sides . The particle is allowed to move freely between and .

• 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 and find the first 6 eigenenergies of the problem in terms of the box side length (), the particle mass () 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 and . Assume that . Representing the enegy eigenstate by and . The Hamiltonian can be written as

a.) We introduce and interaction whose matrix elements, using the above basis vectors, are where v is real. Find the exact values of the energies of the new Hamitonlian,

b.) Assume that . Use time independent perturbation theory to compute the first and second order corrections to the energy levels of when is treated as a perturbation.

c.) Let in above. Assuming the system is initially in the eigenstate of , and the interaction is turned on for a finite time interval . Find the probabitlity that the system will be found in the state at the end of that time interval.