Assignment 11

Problem 1:

Two identical spin zero bosons are placed in a one-dimensional infinite square well,  U(x) = ∞,  x < 0 and x > a, U(x) = 0, 0 < x < a.  The bosons interact weakly with one another via the potential energy function  U(x1,x2) = -aU0δ(x1 - x2), where U0 is a constant with the dimensions of energy.

(a)  First, ignoring the interaction between the particles, find the ground state and first excited state wave functions and the associated energies.
(b)  Use first-order perturbation theory to estimate the effect of the particle-particle interaction on the energies of the ground state and the first excited state.

Problem 2:

Write down the electronic configuration for the Magnesium atom (Z = 12) in its ground state.  Then enumerate the allowed term symbols 2S+1LJ for the ground state from the point of view of angular momentum alone.

Problem 3:

Titanium (Ti) is element No. 22 in the Periodic Table.
(a)  Write down the electronic shell configuration of Ti (i.e., 1s2, 2s2, 2p6, ... ).
(b)  Find the total orbital angular momentum quantum number L, total spin angular momentum quantum number S, and the total angular momentum quantum number J for the ground state of the Ti atom.
(c)  Write down the corresponding spectroscopic term symbol 2S+1XJ  for the ground state, where X represents the orbital angular momentum.
(d)  In a photoemission experiment, an electron is ejected from the 2p shell.  In this process, we will assume that the angular momentum states of the valence electrons remain frozen.  Find the possible values of S, L, and J for the ionized 2p level, ignoring all other levels in the atom.
(e) Couple the ground state orbital angular momentum found in (b) with the one you found in (d).  Do the same for the spin angular momenta in (b) and (d).  What possible values of L and S do you get?
(f ) Find the six possible spectroscopic term symbols for the ionized Ti atom (or final state of the photoemission process).  Just leave the value of J blank, i.e., write them as 2S+1X.
(g)  Find the corresponding multiplicities.

Problem 4:

Consider a system of two non-interacting particles (1, 2) and two orthonormal energy eigenstates (α, β) with energies Eα and Eβ = 3Eα.
Determine the factor by which the probability for finding both particles in the same state for bosons exceeds that for classical particles.

Problem 5:

The Hamiltonian for two interacting spin ½ identical fermions in one dimension is
H = p12/(2m) + p22/(2m) + (mω2/2)(x2 - x1)2.
What is the energy spectrum and what are the corresponding eigenfunctions?
Consider both the motion of the center of mass and the motion about the center of mass.