Problem 1:

In one dimension, the potential energy of an electron as a function of x is given by

U(x) = -30eV exp(-x²/(4Ų)).

Use the variational method to find the energy of the ground state in units of eV.

Problem 2:

Using first order perturbation theory, find the shift in the ground state energy of the one-dimensional harmonic oscillator when the perturbation
(a)  V(x) = bx⁴
(b)  V(x) = bx³
is added to H = ½p²/m + ½mw²x².

Problem 3:

It is known that the stable H^- exists (two electrons bound to a proton).  Estimate the ground state energy of H^- using the variational method.  Assume that each electron moves in a 1s orbit.  Neglect spin.

Useful information:
y_1s(r) = (1/pa₀³)^½exp(-r/a₀)
< y_1s |(1/r)| y_1s > = 1/a₀
< y_1s |Ѳ| y_1s > = -1/a₀²
òd³rd³r’|y_1s(r)|²|y_1s(r’)|²(1/|r-r’|) = 5/(8a₀)

Problem 4:

Solve the perturbed 1-D particle in a box.  Follow the described setup precisely.  The 1-D box is of width 2a centered on x = 0.  The potential V₀ = 0 for  |x| < a and V₀ = ¥ for  |x| > a.
(a)  Write the Hamiltonian and solve for energies and wave functions.  (You must use the symmetric potential form specified.)
(b)  Let there be a small perturbing potential V’ = c - bx² where b = c/a².   Find the first order correction to the energy for the general n-th level.
(c)  Find the first order correction to the unperturbed wave functions.  Hint: In the expansion, non-zero coefficients may be left in integral form.

Problem 5:

Suppose the Hamiltonian of a rigid rotator in a magnetic field is of the form
H = AL² + BL_z + CL_y, if terms quadratic in the field are neglected.
Assuming B >> C, use perturbation theory to lowest non-vanishing order to get appropriate energy eigenvalues.

Problem 6:

(a)  List the 8 possible states of the n = 2 manifold of the hydrogen atom in the common eigenbasis of  L², L_z, S², and S_z, and also in the common eigenbasis of L², S², J², and J_z.
(b)  The 8 states of the n = 2 manifold would be degenerate if not for spin-orbit coupling, and hyperfine splitting.  Let us ignore the hyperfine splitting but treat the spin orbit coupling using first order perturbation theory.  Determine the energies of the 8 states of the n = 2 manifold under the perturbation V = lL×S, where L is the orbital angular momentum operator of the electron and S is the electron's spin operator.
(c)  What is the physical origin of the hyperfine structure?

Problem 7:

A particle of mass m is constrained to move in an infinitely deep, one-dimensional square well extending from -a to +a.  If this particle is under the influence of a perturbation H' = -Ad(x), where A is a constant and d(x) is a delta function at x, calculate the first order corrections to the energy levels.  What is the condition that A must satisfy if the corrected n* energy level is to have a negative energy?