10.8 Problems for This Chapter
585
tems to determine the number of microstates obtained for 3 bosons distributed
over 10 states and for 3 fermions distributed over 10 states. Carry out the same
calculations for 3 bosons (fermions), but now distributed over 100 states and
comment upon any trend that may appear to be developing.
5. To explore briefly the evaluation of the canonical partition function for systems
of quantum particles, we shall consider a simple model that consists of a box
that supports ten single-particle ground-energy states which, for convenience,
we shall set to zero. Determine the number of two-particle states that can
be formed, both when the particles are classical (distinguishable or indistinguishable) and when they are quantum (bosonic or fermionic), and compare
your results with the corresponding value for the canonical partition function.
Determine the probabilities for finding both particles in the same 1-particle
state.
6. If five indistinguishable particles are placed into a container that supports a
set of equally-spaced nondegenerate energy states and supplied with sufficient
energy to form their ground state configuration, what form will that configuration assume for MB (i.e., classical), FD, and BE particles? If one additional unit
of energy is made available for distribution amongst the five particles, how will
these configurations change? What forms will the relevant configurations take
upon supplying another one, two, additional units of energy to the system? How
might we distinguish between systems of indistinguishable and distinguishable
particles?
7. The system internal energy U and average number of particles N are given in
terms of the mean occupation numbers n k defined in Eq. (10.2.22) for Fermi–
Dirac (FD) and Bose–Einstein (BE) systems by Eqs. (10.2.23) and (10.2.21),
respectively. Expression (10.2.22) for the n k is also commonly rewritten in the
convenient form n k = [e β(( k −μ) ± 1] −1 . Employ Eqs. (10.2.21) and (10.2.23)
for N and U to aid in interpreting the meanings of T and μ for systems
consisting of quantum mechanical particles treated in principle in terms of
microcanonical, canonical, or grand ensembles.
8. Consider a system of fermions at temperature T = 77.4 K, (i.e., liquid nitrogen
temperature). Compute the probability for occupation of single-particle states
lying at energies 1 eV, 0.1 eV, 0.02 eV, and 0.01 eV above the value of the
chemical potential μ and, similarly, lying at energies −0.01 eV, −0.02 eV,
−0.1 eV, and −1.0 eV below μ. Compare the results obtained for T = 77.4 K
with those obtained for T = 500 K. Given that the chemical potential for
electrons in metals is typically of the order of 500 eV, what does this behaviour
say about the nature of the Fermi–Dirac distribution for electrons in metals?
9. Consider two single-particle fermion states h, k such that state h lies below
the chemical potential μ by an amount a and state k lies above the chemical
potential by the same amount, i.e., h = μ − a and k = μ + a. How are the
probabilities n k and n k for states h and k related?
10. For a system of bosons at the temperature of liquid nitrogen (i.e., T = 77.4 K),
determine the average occupancy for a single-particle state having energy and
obtain an expression giving the probability for this bosonic state to contain 0, 1,
585
tems to determine the number of microstates obtained for 3 bosons distributed
over 10 states and for 3 fermions distributed over 10 states. Carry out the same
calculations for 3 bosons (fermions), but now distributed over 100 states and
comment upon any trend that may appear to be developing.
5. To explore briefly the evaluation of the canonical partition function for systems
of quantum particles, we shall consider a simple model that consists of a box
that supports ten single-particle ground-energy states which, for convenience,
we shall set to zero. Determine the number of two-particle states that can
be formed, both when the particles are classical (distinguishable or indistinguishable) and when they are quantum (bosonic or fermionic), and compare
your results with the corresponding value for the canonical partition function.
Determine the probabilities for finding both particles in the same 1-particle
state.
6. If five indistinguishable particles are placed into a container that supports a
set of equally-spaced nondegenerate energy states and supplied with sufficient
energy to form their ground state configuration, what form will that configuration assume for MB (i.e., classical), FD, and BE particles? If one additional unit
of energy is made available for distribution amongst the five particles, how will
these configurations change? What forms will the relevant configurations take
upon supplying another one, two, additional units of energy to the system? How
might we distinguish between systems of indistinguishable and distinguishable
particles?
7. The system internal energy U and average number of particles N are given in
terms of the mean occupation numbers n k defined in Eq. (10.2.22) for Fermi–
Dirac (FD) and Bose–Einstein (BE) systems by Eqs. (10.2.23) and (10.2.21),
respectively. Expression (10.2.22) for the n k is also commonly rewritten in the
convenient form n k = [e β(( k −μ) ± 1] −1 . Employ Eqs. (10.2.21) and (10.2.23)
for N and U to aid in interpreting the meanings of T and μ for systems
consisting of quantum mechanical particles treated in principle in terms of
microcanonical, canonical, or grand ensembles.
8. Consider a system of fermions at temperature T = 77.4 K, (i.e., liquid nitrogen
temperature). Compute the probability for occupation of single-particle states
lying at energies 1 eV, 0.1 eV, 0.02 eV, and 0.01 eV above the value of the
chemical potential μ and, similarly, lying at energies −0.01 eV, −0.02 eV,
−0.1 eV, and −1.0 eV below μ. Compare the results obtained for T = 77.4 K
with those obtained for T = 500 K. Given that the chemical potential for
electrons in metals is typically of the order of 500 eV, what does this behaviour
say about the nature of the Fermi–Dirac distribution for electrons in metals?
9. Consider two single-particle fermion states h, k such that state h lies below
the chemical potential μ by an amount a and state k lies above the chemical
potential by the same amount, i.e., h = μ − a and k = μ + a. How are the
probabilities n k and n k for states h and k related?
10. For a system of bosons at the temperature of liquid nitrogen (i.e., T = 77.4 K),
determine the average occupancy for a single-particle state having energy and
obtain an expression giving the probability for this bosonic state to contain 0, 1,