I am trying to figure out how people use the __LINE__ macro at work. How do you guys use it?


Question: What is the the Ehrenfest-Oppenheimer rule on the statistics of composite systems?
Answer: An SE user was kind enough to send me a PDF of the paper, so I'll try to give a summary of my own understanding of it.

The historical context is interesting. This was before the neutron was discovered experimentally, so they thought nuclei were made out of protons and electrons. What we would today describe as a nucleus with mass number $A$ and atomic number $Z$, they would describe as a system consisting of $m$ protons plus $n$ electrons bound inside the nucleus, where $m=A$ and $n=A-Z$. (These $n$ electrons are in addition to the $m$ electrons that are outside the nucleus.) This model gets the statistics of a nucleus wrong when $N$ is odd, and the discrepancy had shown up in the spectroscopy of rotational bands of symmetric diatomic molecules such as $\text{N}_2$.

Regardless of whether you use the archaic model of the nucleus or a modern one, you're dealing with two species of fermions inside the nucleus. This leads to a lot of complications in the notation and equations, which I think is not really crucial to the content of what people would today describe as the Ehrenfest-Oppenheimer rule. I think all the interesting issues can be seen when there is only one species of fundamental fermion present, so that simplified version is what I'll outline here.

The argument, if I'm understanding it properly, seems to be as follows. You have two identical composite systems, which if not interacting would be described by internal quantum numbers $\sigma$ and $\tau$ and quantum numbers $s$ and $t$ describing their center-of-mass motions. Under interchange of the members of the two clusters, the wavefunction $|F\rangle=|st,\sigma\tau\rangle$ representing them has to pick up a phase $\theta=(-1)^k$. We can make such a wavefunction out of a Slater determinant. We then fairly trivially obtain --

The Ehrenfest-Oppenheimer rule: If the internal states of the two clusters are the same, $\sigma=\tau$, then under interchange of the centers of mass $s$ and $t$, the total wavefunction picks up a phase $\theta$.

If we now turn on an interaction between the two clusters, the E-O rule is no longer exact, and the meat of the paper is to show the conditions under which it is a sufficient approximation. The wavefunction F is no longer a stationary state. However, the set of such wavefunctions still represents a complete basis, so we can write the actual physical state $|\phi\rangle$ as a superposition of the F's. This mixes the F's, and the mixing destroys the symmetry under interchange of $s$ and $t$. However, the mixing depends on matrix elements of the form

$$ \langle st,\sigma\tau | H | s't',\sigma'\tau' \rangle $$

where $\sigma\ne\sigma'$, $\tau\ne\tau'$, i.e., the internal states are different. The resulting mixing is then small when the energy scale of the interaction between the two clusters is small compared to the energy differences $E_\sigma-E_{\sigma'}$ for $\sigma\ne\sigma'$. This is, for example, an excellent approximation for the nuclei in a diatomic molecule.