Chapter 10
Quantum Statistics
This chapter examines the statistics imposed by quantum mechanical symmetry
restrictions upon wavefunctions for systems of two types of inherently indistinguishable quantum mechanical particles, called bosons and fermions. The distinctions
between fermions and bosons, symmetry requirements that apply to them, and the
natures of their statistical descriptions are elaborated using the grand ensemble.
Each type of quantum statistics reduces under appropriate conditions to classical
statistics. The 3 He isotope, conduction electrons in metals, and free electrons in
semiconductors are treated as ideal fermion gases, while the 4 He isotope and
photons are treated as ideal boson gases. Discussions of strongly degenerate fermion
and boson gases are presented. Both ideal fermion and ideal boson gases have
nonzero second virial coefficients at low temperatures. The chemical potentials for
fermion and boson gases show distinctly different temperature dependences at low
temperatures, but merge with the classical mechanical chemical potential at high
temperatures.
10.1 Setting the Stage
We have thus far considered systems in which we have not had to worry at a fundamental level about quantum mechanical symmetry restrictions on the wavefunctions
for a molecular system, and have consequently restricted our considerations to cases
in which the number of available states by far exceeds the number of particles. In
this way situations in which more than one particle is to be found in a given state
are extremely rare, i.e., on average only one particle occupies a particular quantum
mechanical state of the N -particle system. At a more fundamental level, however,
we should take into consideration the fact that if we are dealing with fermions
(particles with an intrinsic half-odd-integer spin), the N -body wavefunction must
be antisymmetric to the interchange of two identical fermions, while for bosons
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_10
511
Quantum Statistics
This chapter examines the statistics imposed by quantum mechanical symmetry
restrictions upon wavefunctions for systems of two types of inherently indistinguishable quantum mechanical particles, called bosons and fermions. The distinctions
between fermions and bosons, symmetry requirements that apply to them, and the
natures of their statistical descriptions are elaborated using the grand ensemble.
Each type of quantum statistics reduces under appropriate conditions to classical
statistics. The 3 He isotope, conduction electrons in metals, and free electrons in
semiconductors are treated as ideal fermion gases, while the 4 He isotope and
photons are treated as ideal boson gases. Discussions of strongly degenerate fermion
and boson gases are presented. Both ideal fermion and ideal boson gases have
nonzero second virial coefficients at low temperatures. The chemical potentials for
fermion and boson gases show distinctly different temperature dependences at low
temperatures, but merge with the classical mechanical chemical potential at high
temperatures.
10.1 Setting the Stage
We have thus far considered systems in which we have not had to worry at a fundamental level about quantum mechanical symmetry restrictions on the wavefunctions
for a molecular system, and have consequently restricted our considerations to cases
in which the number of available states by far exceeds the number of particles. In
this way situations in which more than one particle is to be found in a given state
are extremely rare, i.e., on average only one particle occupies a particular quantum
mechanical state of the N -particle system. At a more fundamental level, however,
we should take into consideration the fact that if we are dealing with fermions
(particles with an intrinsic half-odd-integer spin), the N -body wavefunction must
be antisymmetric to the interchange of two identical fermions, while for bosons
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_10
511