588
10 Quantum Statistics
24. Obtain an expression for the internal energy, U , for spin-I a fermions in a twodimensional square potential box, and show that the internal energy per fermion
is one-half the Fermi energy.
25. Show that the density of states, FD ((), for particles in a 2D square potential
box is constant, and express it in terms of N and F .
26. Consider an electron gas confined to a 2D square potential box. Employ the
constancy of the density of states for particles in such a 2D potential box
(as found in Problem 25) to obtain an expression for N for the electron gas,
and show that the chemical potential for such an electron gas is given by
μ = k B T ln(e ββ F − 1).
27. For temperatures T such that k B T F , compare the expression obtained
in Problem 26 for the chemical potential of a two-dimensional electron-gas
with the expression for the chemical potential μ Eq. (5.1.8) for the translational
states of a classical gas and comment.
28. Obtain an expression for the heat capacity, C V (T ), for spin-I a fermions
confined to a two-dimensional square potential box.
29. Determine the efficiency, η photon gas , of a Carnot engine for which the working
substance is a thermalized photon quantum gas (i.e., blackbody radiation) and
compare your expression for η photon gas with that for an ideal classical gas.
30. Set up physical descriptions to illustrate free expansions of classical and
quantum ideal (blackbody radiation) gases and utilize them to compare and
contrast the two free expansions.
References
1. P.T. Landsberg, J. Dunning-Davies, D. Pollard, Amer. J. Phys. 62, 712 (1994)
2. R.A. Aziz, C.C.K. Wong, F.R.W. McCourt, Mol. Phys. 61, 1487 (1987)
3. G. Cook, R.H. Dickerson, Amer. J. Phys. 63, 737 (1995)
4. V.C. Aguilera-Nevarro, G.A. Estevez, Amer. J. Phys. 56, 456 (1988)
5. W.M. Haynes (ed.), Handbook of Physics and Chemistry, 97th edn. (CRC Press, Baton Rouge,
2016)
6. W.S. Corak, M.P. Garfunkel, C.B. Satterthwaite, A. Wexler, Phys. Rev. 98, 1699 (1955)
7. R. Kubo, H. Ichimura, T. Usai, N. Hashizume, Statistical Mechanics. An Advanced Course
with Problems and Solutions (North-Holland, Amsterdam, 1965)
8. R.K. Pathria, Statistical Mechanics (Pergamon, London, 1972)
9. H.S. Leff, Amer. J. Phys. 83, 362 (2015)
10. B.H. Bransden, C.J. Joachain, Quantum Mechanics, , Chap. 14, 2nd edn. (Prentice Hall,
London, 2000)
11. W. Greiner, J. Neise, H. Stöker, Thermodynamics and Statistical Mechanics, Chap. 10
(Springer, New York, 1995)
10 Quantum Statistics
24. Obtain an expression for the internal energy, U , for spin-I a fermions in a twodimensional square potential box, and show that the internal energy per fermion
is one-half the Fermi energy.
25. Show that the density of states, FD ((), for particles in a 2D square potential
box is constant, and express it in terms of N and F .
26. Consider an electron gas confined to a 2D square potential box. Employ the
constancy of the density of states for particles in such a 2D potential box
(as found in Problem 25) to obtain an expression for N for the electron gas,
and show that the chemical potential for such an electron gas is given by
μ = k B T ln(e ββ F − 1).
27. For temperatures T such that k B T F , compare the expression obtained
in Problem 26 for the chemical potential of a two-dimensional electron-gas
with the expression for the chemical potential μ Eq. (5.1.8) for the translational
states of a classical gas and comment.
28. Obtain an expression for the heat capacity, C V (T ), for spin-I a fermions
confined to a two-dimensional square potential box.
29. Determine the efficiency, η photon gas , of a Carnot engine for which the working
substance is a thermalized photon quantum gas (i.e., blackbody radiation) and
compare your expression for η photon gas with that for an ideal classical gas.
30. Set up physical descriptions to illustrate free expansions of classical and
quantum ideal (blackbody radiation) gases and utilize them to compare and
contrast the two free expansions.
References
1. P.T. Landsberg, J. Dunning-Davies, D. Pollard, Amer. J. Phys. 62, 712 (1994)
2. R.A. Aziz, C.C.K. Wong, F.R.W. McCourt, Mol. Phys. 61, 1487 (1987)
3. G. Cook, R.H. Dickerson, Amer. J. Phys. 63, 737 (1995)
4. V.C. Aguilera-Nevarro, G.A. Estevez, Amer. J. Phys. 56, 456 (1988)
5. W.M. Haynes (ed.), Handbook of Physics and Chemistry, 97th edn. (CRC Press, Baton Rouge,
2016)
6. W.S. Corak, M.P. Garfunkel, C.B. Satterthwaite, A. Wexler, Phys. Rev. 98, 1699 (1955)
7. R. Kubo, H. Ichimura, T. Usai, N. Hashizume, Statistical Mechanics. An Advanced Course
with Problems and Solutions (North-Holland, Amsterdam, 1965)
8. R.K. Pathria, Statistical Mechanics (Pergamon, London, 1972)
9. H.S. Leff, Amer. J. Phys. 83, 362 (2015)
10. B.H. Bransden, C.J. Joachain, Quantum Mechanics, , Chap. 14, 2nd edn. (Prentice Hall,
London, 2000)
11. W. Greiner, J. Neise, H. Stöker, Thermodynamics and Statistical Mechanics, Chap. 10
(Springer, New York, 1995)