11.2 Single-Particle versus System States

The purpose of this section is to describe the generic form of the energy eigen­functions of a system of weakly inter­acting particles.

The total number of particles will be indicated by $I$. If the inter­actions between the $I$ particles are ignored, any energy eigen­function of the complete system of $I$ particles can be written in terms of single-particle energy eigen­functions $\pp1/{\skew0\vec r}//z/,\pp2/{\skew0\vec r}//z/,\ldots$.

The basic case is that of non­interacting particles in a box, like discussed in chapter 6.2. For such particles the single-particle eigen­functions take the spatial form

$$\pp{n}//// = \sqrt{\frac{8}{\ell_x\ell_y\ell_z}} \sin(k_xx)\sin(k_yy)\sin(k_zz)$$

where $k_x$, $k_y$, and $k_z$ are constants, called the “wave number components.” Different values for these constants corre­spond to different single-particle eigen­functions, with single-particle energy

$${\vphantom' E}^{\rm p}_n = \frac{\hbar^2}{2m} (k_x^2 + k_y^2 + k_z^2) = \frac{\hbar^2}{2m} k^2$$

The single-particle energy eigen­functions will in this chapter be numbered as $n$ $\vphantom0\raisebox{1.5pt}{$=$}$ $1,2,3,\ldots,N$. Higher values of index $n$ corre­spond to eigen­functions of equal or higher energy ${\vphantom' E}^{\rm p}_n$.

The single-particle eigen­functions do not always corre­spond to a particle in a box. For example, particles caught in a magnetic trap, like in the Bose-Einstein condens­ation experi­ments of 1995, might be better described using harmonic oscillator eigen­functions. Or the particles might be restricted to move in a lower-di­mensional space. But a lot of the formulae you can find in literature and in this chapter are in fact derived assuming the simplest case of non­interacting particles in a roomy box.

The details of the single-particle energy eigen­functions are not really that important in this chapter. What is more inter­esting are the energy eigen­functions $\psi^{\rm S}_q$ of complete systems of particles. It will be assumed that these system eigen­functions are numbered using a counter $q$, but the way they are numbered also does not really make a difference to the analysis.

As long as the inter­actions between the particles are weak, energy eigen­functions of the complete system can be found as products of the single-particle ones. As an important example, at absolute zero temperature, all particles will be in the single-particle ground state $\pp1////$, and the system will be in its ground state

$$\psi^{\rm S}_1 = \pp1/{\skew0\vec r}_1//z1/ \pp1/{\skew0... ... \pp1/{\skew0\vec r}_5//z5/ \ldots \pp1/{\skew0\vec r}_I//zI/$$

where $I$ is the total number of particles in the system. This does assume that the single-particle ground state energy ${\vphantom' E}^{\rm p}_1$ is not degenerate. More importantly, it assumes that the $I$ particles are not identical fermions. According to the exclusion principle, at most one fermion can go into a single-particle state. (For spin $\leavevmode\kern.03em \raise.7ex\hbox{\the\scriptfont0 1}\kern-.2em /\kern-.2em\lower.4ex\hbox{\the\scriptfont0 2}\kern.05em$ fermions like electrons, two can go into a single spatial state, one in the spin-up version, and the other in the spin-down one.)

Statistical thermo­dynamics, in any case, is much more inter­ested in temperatures that are not zero. Then the system will not be in the ground state, but in some combin­ation of system eigen­functions of higher energy. As a completely arbitrary example of such a system eigen­function, take the following one, describing $I$ $\vphantom0\raisebox{1.5pt}{$=$}$ 36 different particles:

$$\psi^{\rm S}_q = \pp24/{\skew0\vec r}_1//z1/ \pp4/{\skew... .../{\skew0\vec r}_5//z5/ \ldots \pp54/{\skew0\vec r}_{36}//z36/$$

This system eigen­function has an energy that is the sum of the 36 single-particle eigenstate energies involved:

$${\vphantom' E}^{\rm S}_q = {\vphantom' E}^{\rm p}_{24} + {... ...vphantom' E}^{\rm p}_6 + \ldots + {\vphantom' E}^{\rm p}_{54}$$

Figure 11.1: Graphical depiction of an arbitrary system energy eigen­function for 36 distin­guishable particles.

To understand the arguments in this chapter, it is essential to visualize the system energy eigen­functions as in figure 11.1. In this figure the single-particle states are shown as boxes, and the particles that are in those particular single-particle states are shown inside the boxes. In the example, particle 1 is inside the $\pp24////$ box, particle 2 is inside the $\pp4////$ one, etcetera. It is just the reverse from the mathematical expression above: the mathematical expression shows for each particle in turn what the single-particle eigenstate of that particle is. The figure shows for each type of single-particle eigenstate in turn what particles are in that eigenstate.

To simplify the analysis, in the figure single-particle eigen­states of about the same energy have been grouped together on “shelves.” (As a consequence, a subscript to a single-particle energy ${\vphantom' E}^{\rm p}$ may refer to either a single-particle eigen­function number $n$ or to a shelf number $s$, depending on context.) The number of single-particle states on a shelf is intended to roughly simulate the density of states of the particles in a box as described in chapter 6.3. The larger the energy, the more single-particle states there are at that energy; it increases like the square root of the energy. This may not be true for other situations, such as when the particles are confined to a lower-di­mensional space, compare chapter 6.12. Various formulae given here and in literature may need to be adjusted then.

Of course, in normal non­nano appli­cations, the number of particles will be astronomi­cally larger than 36 particles; the example is just a small illustr­ation. Even a millimol of particles means on the order of 10$\POW9,{20}$ particles. And unless the temperature is incredibly low, those particles will extend to many more single-particle states than the few shown in the figure.

Next, note that you are not going to have something like 10$\POW9,{20}$ different types of particles. Instead they are more likely to all be helium atoms, or all electrons or so. If their wave functions overlap non­trivially, that makes a big difference because of the symmetr­ization requirements of the system wave function.

Consider first the case that the $I$ particles are all identical bosons, like plain helium atoms. In that case the wave function must be symmetric, unchanged, under the exchange of any two of the bosons, and the example wave function above is not. If, for example, particles 2 and 5 are exchanged, it turns the example wave function from

$$\psi^{\rm S}_q = \pp24/{\skew0\vec r}_1//z1/ \pp4/{\skew... .../{\skew0\vec r}_5//z5/ \ldots \pp54/{\skew0\vec r}_{36}//z36/$$

into

$$\psi^{\rm S}_{{\underline q}} = \pp24/{\skew0\vec r}_1//... .../{\skew0\vec r}_5//z5/ \ldots \pp54/{\skew0\vec r}_{36}//z36/$$

and that is simply a different wave function, because the states are different, independent functions. In terms of the pictorial represen­tation figure 11.1, swapping the numbers “2” and “5” in the particles changes the picture.

Figure 11.2: Graphical depiction of an arbitrary system energy eigen­function for 36 identical bosons.

As chapter 5.7 explained, to eliminate the problem that exchanging particles 2 and 5 changes the wave function, the original and exchanged wave functions must be combined together. And to eliminate the problem for any two particles, all wave functions that can be obtained by merely swapping numbers must be combined together equally into a single wave function multi­plied by a single undetermined coefficient. In terms of figure 11.1, we need to combine the wave functions with all possible permu­tations of the numbers inside the particles into one. And if all permu­tations of the numbers are equally included, then those numbers no longer add any non­trivial additional infor­mation; they may as well be left out. That makes the pictorial represen­tation of an example system wave function for identical bosons as shown in figure 11.2.

Figure 11.3: Graphical depiction of an arbitrary system energy eigen­function for 33 identical fermions.

For identical fermions, the situation is similar, except that the different wave functions must be combined with equal or opposite sign, depending on whether it takes an odd or even number of particle swaps to turn one into the other. And such wave functions only exist if the $I$ single-particle wave functions involved are all different. That is the Pauli exclusion principle. The pictorial represen­tation figure 11.2 for bosons is totally unacceptable for fermions since it uses many of the single-particle states for more than one particle. There can be at most one fermion in each type of single-particle state. An example of a wave function that is acceptable for a system of identical fermions is shown in figure 11.3.

Looking at the example pictorial represen­tations for systems of bosons and fermions, it may not be surprising that such particles are often called “indistin­guishable.“ Of course, in classical quantum mechanics, there is still an electron 1, an electron 2, etcetera; they are mathemati­cally distin­guished. Still, it is convenient to use the term “distin­guishable” for particles for which the symmetr­ization requirements can be ignored.

The prime example is the atoms of an ideal gas in a box; almost by definition, the inter­actions between such atoms are negligible. And that allows the quantum results to be referred back to the well-understood properties of ideal gases obtained in classical physics. Probably you would like to see all results follow naturally from quantum mechanics, not classical physics, and that would be very nice indeed. But it would be very hard to follow up on. As Baierlein [4, p. 109] notes, real-life physics adopts whichever theoretical approach offers the easiest calcul­ation or the most insight. This book’s approach really is to formulate as much as possible in terms of the quantum-mechanical ideas discussed here. But do be aware that it is a much more messy world when you go out there.