Page 239 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

230 Systems of identical particles

lithium-7 atoms 4,5 have also been cooled suf®ciently to undergo Bose±Einstein
condensation.
Although we have explained Bose±Einstein condensation as a characteristic
of an ideal or nearly ideal gas, i.e., a system of non-interacting or weakly
interacting particles, systems of strongly interacting bosons also undergo
similar transitions. Liquid helium-4, as an example, has a phase transition at
2.18 K and below that temperature exhibits very unusual behavior. The proper-
ties of helium-4 at and near this phase transition correlate with those of an ideal
Bose±Einstein gas at and near its condensation temperature. Although the
actual behavior of helium-4 is due to a combination of the effects of quantum
statistics and interparticle forces, its qualitative behavior is related to Bose±
Einstein condensation.

Problems
^
^
8.1 Show that the exchange operators P in equation (8.4) and P áâ in (8.20) are
hermitian.
8.2 Noting from equation (8.10) that
Ø(1, 2) ˆ 2 ÿ1=2 (Ø S ‡ Ø A )
Ø(2, 1) ˆ 2 ÿ1=2 (Ø S ÿ Ø A )
show that Ø(1, 2) and Ø(2, 1) are orthogonal if Ø S and Ø A are normalized.
8.3 Verify the validity of the relationships in equation (8.19).
8.4 Verify the validity of the relationships in equation (8.22).
8.5 Apply equation (8.12) to show that Ø S and Ø A in (8.26) are normalized.
8.6 Consider two identical non-interacting particles, each of mass m, in a one-
dimensional box of length a. Suppose that they are in the same spin state so that
spin may be ignored.
(a) What are the four lowest energy levels, their degeneracies, and their corre-
sponding wave functions if the two particles are distinguishable?
(b) What are the four lowest energy levels, their degeneracies, and their corre-
sponding wave functions if the two particles are identical fermions?
(c) What are the four lowest energy levels, their degeneracies, and their corre-
sponding wave functions if the two particles are identical bosons?
8.7 Consider a crude approximation to the ground state of the lithium atom in which
the electron±electron repulsions are neglected. Construct the ground-state wave
function in terms of the hydrogen-like atomic orbitals.

4 C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet (1995) Phys. Rev. Lett. 75, 1687.
5 C. C. Bradley, C. A. Sackett, and R. G. Hulet (1997) Phys. Rev. Lett. 78, 985.