Abstract
The chapter illustrates the properties of many-particle systems. The quantum-mechanical description of the latter is obtained by solving the time-dependent Schrödinger equation. After commenting the simplifications that occur when the Hamiltonian operator is separable, the important issue of the symmetry or antisymmetry of the wave function is introduced, to the purpose of illustrating the peculiar properties possessed by the systems of identical particles. Then, the concept of spin and the exclusion principle are introduced. After a general discussion, the above concepts are applied to the important case of a conservative system, and further properties related to the separability of the Hamiltonian operator are worked out. The remaining part of the chapter is devoted to the derivation of the equilibrium statistics in the quantum case (Fermi-Dirac and Bose-Einstein statistics). The connection between the microscopic statistical concepts and the macroscopic thermodynamic properties is illustrated in the complements, where two important examples of calculation of the density of states are also given.
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Notes
- 1.
By way of example, (15.3) is separable if \(V = V_1\Big(\mathbf{r}_1,t\Big) + V_2\Big(\mathbf{r}_2,t\Big)\).
- 2.
A “group” of coordinates may also consist of a single coordinate.
- 3.
The reasoning outlined here does not apply to systems where the particles are different: they can be distinguished in \(\vert \psi \vert^2\), e.g., by the mass or electric charge.
- 4.
The names “bosons”, “fermions” of the two subclasses have this origin: when a system of identical particles is in thermodynamic equilibrium, the particles’ energy follows a statistical distribution whose expression is named after Bose and Einstein (Sect. 15.8.2) and, respectively, Fermi and Dirac (Sect. 15.8.1).
- 5.
It must be noted, however, that in condensed-matter physics two electrons or other fermions may bind together at low temperatures to form a so-called Cooper pair, which turns out to have an integer spin, namely, it is a composite boson [20].
- 6.
Like the Heisenberg principle illustrated in Sect. {10.6}, that of Pauli was originally deduced from heuristic arguments. The analysis of this section shows in fact that it is a theorem rather than a principle.
- 7.
The use of energy intervals does not entail a loss of generality, as the subsequent treatment of the general case will show.
- 8.
Here the eigenvalues of the Hamiltonian operator are discrete because the system is enclosed in a container, hence the wave function is normalizable. As usual, the notation n(i) stands for a group of indices.
- 9.
The units are \([g],[N] = \mbox{J}^{-1}\).
- 10.
Compare with the non-equilibrium definition of entropy introduced in Sect. {6.6.3} and the note therein.
- 11.
The geometrical configuration is kept similar to the original one during the change in volume.
- 12.
Here the term “volume” is used in a broader meaning; in fact, the units of \(\pi^3 / V\) are \(\mathrm{m^{-3}}\).
- 13.
As noted above the \(\mathbf{k}\) vectors, hence the values of energy corresponding to them, are distributed very densely. This makes it possible to treat E as a continuous variable.
- 14.
The calculation shown here is equivalent to counting the number of elements of volume \((2\,\pi)^3 /\Big(d_1\,d_2\,d_3\Big)\) that belong to the spherical shell drawn in Fig. 15.4. The result is then multiplied by 2 to account for spin.
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Problems
Problems
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15.1
Estimate the extension of the energy region where the main variation of the Fermi statistics (15.49) occurs.
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Rudan, M. (2015). Many-Particle Systems. In: Physics of Semiconductor Devices. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1151-6_15
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DOI: https://doi.org/10.1007/978-1-4939-1151-6_15
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