Stationary perturbation theory, degenerate states

Problem:

Consider a two-dimensional infinite potential square well of width L,
(U = 0 for 0 < x, y < L, U = infinite everywhere else) with an added perturbation

H' = g sin(2πx/L)sin(2πy/L).

(a)  Calculate the first order perturbation to the ground state energy eigenvalue.
(b)  Calculate the first order perturbation to the first excited state energy eigenvalue.

Solution:

Problem:

A particle of mass m is trapped in a 3-dimensional, infinite square well.
V(x,y,z) = 0, if x, y, and z are less than a and not negative,
V(x,y,z) = ∞, otherwise.
(a)  What are the two lowest energy eigenvalues?  Are these energies degenerate?
(b)  The box is placed into a uniform gravitational field with gravitational acceleration g in the negative z-direction.  Use perturbation theory to calculate first-order corrections to the energy eigenvalues from part (a).

Solution:

Problem:

Consider a charged particle on a ring of unit radius with flux Φ/Φ0 = a passing through the ring, where Φ0 = h/(2e) is the flux quantum.  The Hamiltonian operator can be written as H = H0 + U, where
H0 = (i∂/∂θ + α)2  and U = U0 cosθ.
θ is the angular coordinate.  We have chosen units with ħ = 2m = 1.
(a)  Find the complete set of eigenvalues and eigenfunctions of H0.
(b)  Use perturbation theory to find the first and second-order corrections to the ground state energy E0 of H0 due to the perturbation U for 0 < α < ½ .
(c)  For α = ½ the ground state energy of H0 is degenerate.  Find the first-order correction to E0 for this case.

Solution:

Problem:

Calculate the splitting induced among the degenerate n = 2 levels of a hydrogenic atom, when this atom is placed in a uniform electric field E pointing in the z-direction.  This is the linear Stark effect.  You may use the following explicit hydrogenic wave functions |nlm>.

|200> = (4π)(2a)-3/2(2 - r/a)exp(-r/2a),
|211> = (8π)(2a)-3/2(r/a) exp(-r/2a) sinθ e,
|210> = (4π)(2a)-3/2(r/a) exp(-r/2a) cosθ,
|21-1> = (8π)(2a)-3/2(r/a) exp(-r/2a) sinθ e-iφ.

0e-br rn dr = n!/bn+1.
Hint: Exploit symmetries!

Solution:

Problem:

Positronium is a bound state of an electron and a positron.
(a)  What is the energy eigenvalue for the 1s state?
(b)  If the spin-spin interaction is neglected, what is the degree of degeneracy of the ground state?
(c)  Assume that the spin-spin interaction is given by ASeSp, where Se and Sp are the spin operators for the electron and positron respectively, and A is a coupling constant.  Show that the ground state is split into two states and find the degree of degeneracy of those states.

Solution:

Problem:

A valence electron in an alkali atom is in a p-orbital (l = 1).  Consider the simultaneous interactions of an external magnetic field B = Bk and the spin-orbit interaction.  The two interactions are described by the potential energy
U = (A/ħ2)LS – (μB/ħ)(L + 2S)∙B.
(a)  Describe the energy levels of this l = 1 electron for B = 0.
(b)  Describe the energy levels of this l = 1 electron for weak magnetic fields.  Use the projection theorem. (This is the Zeeman effect.)
(c)  Describe the energy levels for strong magnetic fields so that the spin-orbit term in U can be ignored.  (This is called the Paschen-Back effect.)
Projection theorem:  Inside a subspace E(k,j) the matrix elements of the component of a vector observable V are proportional to the matrix elements of the corresponding component of J
Therefore <k,j,mj'|(L+2S)|k,j,mj> = [<(L+2S)J>kj/(j(j + 1)ħ2)] <k,j,mj'|J|k,j,mj>.

Solution: