6.1 Intro to Particles in a Box

Since most macroscopic systems are very hard to analyze in quantum-mechanics, simple systems are very important. They allow insight to be achieved that would be hard to obtain otherwise. One of the simplest and most important systems is that of multiple noninteracting particles in a box. For example, it is a starting point for quantum thermodynamics and the quantum description of solids.

It will be assumed that the particles do not interact with each other, nor with anything else in the box. That is a dubious assumption; interactions between particles are essential to achieve statistical equilibrium in thermodynamics. And in solids, interaction with the atomic structure is needed to explain the differences between electrical conductors, semiconductors, and insulators. However, in the box model such effects can be treated as a perturbation. That perturbation is ignored to leading order.

In the absence of interactions between the particles, the possible quantum states, or energy eigenfunctions, of the complete system take a relatively simple form. They turn out to be products of single particle energy eigenfunctions. A generic energy eigenfunction for a system of $I$ particles is:

 $\displaystyle {\psi^{\rm S}_{{\vec n}_1,{\vec n}_2,\ldots,{\vec n}_i,\ldots,{\v...
...}_2, S_{z2},\ldots,{\skew0\vec r}_i, S_{zi},\ldots,{\skew0\vec r}_I, S_{zI}) =}$
     $\displaystyle \pp{\vec n}_1/{\skew0\vec r}_1//z1/ \times
\pp{\vec n}_2/{\skew0\...
...vec r}_i//zi/ \times
\ldots \times
\pp{\vec n}_I/{\skew0\vec r}_I//zI/\qquad%
$  (6.1)

In such a system eigenfunction, particle number $i$ out of $I$ is in a single-particle energy eigenfunction $\pp{\vec n}_i/{\skew0\vec r}_i//zi/$. Here ${\skew0\vec r}_i$ is the position vector of the particle, and $S_{zi}$ its spin in a chosen $z$-​direction. The subscript ${\vec n}_i$ stands for whatever quantum numbers characterize the single-particle eigenfunction. A system wave function of the form above, a simple product of single-particles ones, is called a “Hartree product.”

For noninteracting particles confined inside a box, the single-particle energy eigenfunctions, or single-particle states, are essentially the same ones as those derived in chapter 3.5 for a particle in a pipe with a rectangular cross section. However, to account for nonzero particle spin, a spin-dependent factor must be added. In any case, this chapter will not really be concerned that much with the detailed form of the single-particle energy states. The main quantities of interest are their quantum numbers and their energies. Each possible set of quantum numbers will be graphically represented as a point in a so-called “wave number space.” The single-particle energy is found to be related to how far that point is away from the origin in that wave number space.

For the complete system of $I$ particles, the most interesting physics has to do with the (anti) symmetrization requirements. In particular, for a system of identical fermions, the Pauli exclusion principle says that there can be at most one fermion in a given single-particle state. That implies that in the above Hartree product each set of quantum numbers ${\vec n}$ must be different from all the others. In other words, any system wave function for a system of $I$ fermions must involve at least $I$ different single-particle states. For a macroscopic number of fermions, that puts a tremendous restriction on the wave function. The most important example of a system of identical fermions is a system of electrons, but systems of protons and of neutrons appear in the description of atomic nuclei.

The antisymmetrization requirement is really more subtle than the Pauli principle implies. And the symmetrization requirements for bosons like photons or helium-4 atoms are nontrivial too. This was discussed earlier in chapter 5.7. Simple Hartree product energy eigenfunctions of the form (6.1) above are not acceptable by themselves; they must be combined with others with the same single-particle states, but with the particles shuffled around between the states. Or rather, because shuffled around sounds too much like Las Vegas, with the particles exchanged between the states.


Key Points
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Systems of noninteracting particles in a box will be studied.

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Interactions between the particles may have to be included at some later stage.

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System energy eigenfunctions are obtained from products of single-particle energy eigenfunctions.

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(anti) symmetrization requirements further restrict the system energy eigenfunctions.