Description
Problem 1:
Consider a non-interacting one-dimensional non-relativistic spinless quantum gas confined
to a “rod” of length L. Assuming periodic boundary conditions, calculate the density of
states.
Problem 2:
Consider a system consisting of N non-interacting particles each with isospin I = 3/2. The
energies of the states with different projection quantum numbers mI (the eigenvalues of ˆIz
are ¯hmI ;
ˆIz denotes the z-component of ˆ~I) are given by
E(mI = −3/2) = E1, (1)
E(mI = −1/2) = E2, (2)
E(mI = 1/2) = E3, (3)
E(mI = 3/2) = E3, (4)
with
E1 < E2 < E3 (5)
and
∆12 = E2 − E1 ∆23 = E3 − E2. (6)
You may treat the particles as distinguishable throughout.
(a) Without using the partition function, give the value of the total energy hEi at temperatures (i) T = 0, (ii) ∆12 kT ∆23, and (iii) ∆23 T. Provide a justification for your
results. Sketch hEi as a function of temperature.
(b) What is the occupation of the four different mI -states in the T → ∞ limit. Without
using the partition function, give a value of the specific heat in the T → ∞ limit. Provide
a justification for your results.
(c) Without using the partition function, give the value of the average h
ˆIzi of the isospin
z-component per particle at temperatures (i) T = 0, (ii) ∆12 kT ∆23, and (iii)
∆23 T. Provide a justification for your results. Sketch h
ˆIzi as a function of the temperature.
(d) Using the partition function, compute h
ˆIzi in the T → ∞ limit. How is your result
related to the results in part (c)?
Problem 3:
(a) Calculate the density of states D() for a non-interacting three-dimensional gas of spin0 particles confined to a cube of volume L
3 with periodic boundary conditions.
(b) Calculate D() for a non-interacting three-dimensional gas of spin-0 particles confined
to a cube of volume L
3 with hard wall boundary conditions (i.e., boundary conditions such
that the single-particle wave function vanishes at the edges of the box).
Problem 4:
This problem considers a quantum mechanical system that contains three non-interacting
particles. The spatial degree of the particle can be in one of two states (the spatial degree
is for simplicity assumed to be the x-coordinate): ψ1(x) with single-particle energy E1 or
ψ2(x) with single-particle energy E2, where E1 < E2.
In some of the cases considered
below, the particles have a spin s. Assume that the energy levels are independent of the
projection quantum number ms.
For each of the following cases, write down (i) the zero-temperature energy, (ii) the degeneracy, and (iii) the normalized zero-temperature wave function(s) of the three-particle
quantum system.
(a) Three spinless Boltzmann particles.
(b) Three identical spin-0 bosons.
(c) Three identical spin-1/2 fermions.
(d) Three identical spin-3/2 fermions.
