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:

$${\psi^{\rm S}_{{\vec n}_1,{\vec n}_2,\ldots,{\vec n}_i,\ldots,{\v... ...}_2, S_{z2},\ldots,{\skew0\vec r}_i, S_{zi},\ldots,{\skew0\vec r}_I, S_{zI}) =}$$

$$\pp{\vec n}_1/{\skew0\vec r}_1//z1/ \times \pp{\vec n}_2/{\skew... ...r}_i//zi/ \times \ldots \times \pp{\vec n}_I/{\skew0\vec r}_I//zI/\qquad%$$ (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.