# What is the the Ehrenfest-Oppenheimer rule on the statistics of composite systems?

Ehrenfest 1931 gives an argument to the effect that the application of the spin-statistics theorem to composite systems is valid, but only as an approximation and under certain conditions. Unfortunately this article is behind a paywall. The abstract reads as follows:

From Pauli's exclusion principle we derive the rule for the symmetry of the wave functions in the coordinates of the center of gravity of two similar stable clusters of electrons and protons, and justify the assumption that the clusters satisfy the Einstein-Bose or Fermi-Dirac statistics according to whether the number of particles in each cluster is even or odd. The rule is shown to become invalid only when the interaction between the clusters is large enough to disturb their internal motion.

The quaintness of the abstract is enhanced by the mysterious failure to mention neutrons -- this being explained by the fact that the neutron was not discovered experimentally until the following year. Pretty amusing that the American Physical Society still owns the copyright on a paper this old, one of whose authors was probably paid through my grandparents' and great-grandparents' income taxes.

Googling quickly turns up various loose verbal characterizations of this result, without proof. Can anyone supply an outline of how the argument works and possibly a more precise statement of what the result really is?

Some sources seem to refer to this as the Wigner-Ehrenfest-Oppenheimer (WEO) result or Ehrenfest-Oppenheimer-Bethe (EOB) rule. Historically, Feynman seems to have used this result to show that Cooper pairs were bosonic.

Although I've tagged this question as quantum field theory, and the spin-statistics theorem is relativistic, the 1931 date seems to indicate that this would have been a result in nonrelativistic quantum mechanics. This answer by akhmeteli sketches what seems to be a similar relativistic result from a book by Lipkin.

In the limit of low-energy experiments, the EO rule says the usual spin-statistics connection applies, and this seems extremely plausible. If not, then it would be too good to be true: we would be able to probe the composite structure of a system to arbitrarily high energies without having to build particle accelerators.

Bethe 1997 has the following argument:

Consider now tightly bound composite objects like nuclei. It then makes sense to ask for the symmetry of the wavefunction of a system containing many identical objects of the same type, e.g., many He4 nuclei. This symmetry can be deduced by imagining that the interchange of two composites is carried out particle by particle. Each interchange of fermions changes the sign of the wavefunction. Hence the composite will be a fermion if and only if it contains an odd number of fermions[...]

Note the qualifier "tightly bound," which implies a restriction to low-energy experiments. The abstract of Ehrenfest 1931 seems to say that this is a necessary assumption, but it is never used in Bethe and Jackiw's argument, and this suggests that their argument is an oversimplification.

Fujita 2009 recaps the Bethe-Jackiw argument, but then says:

We shall see later that these arguments are incomplete. We note that Feynman used these arguments to deduce that Cooper pairs [5] (pairons) are bosonic [6]. The symmetry of the many-particle wavefunction and the quantum statistics for elementary particles are one-to-one [1]. [...] But no one-to-one correspondence exists for composites since composites, by construction, have extra degrees of freedom. Wavefunctions and second-quantized operators are important auxiliary quantum variables, but they are not observables in Dirac's sense [1]. We must examine the observable occupation numbers for the study of the quantum statistics of composites. In the present chapter we shall show that EOB's rule applies to the Center-of-Mass (CM) motion of composites.

Unfortunately I only have access to Fujita through a keyhole, so I can't see what the references are or any of the details that they promise in this beginning-of-chapter preview.

Bethe and Jackiw, Intermediate Quantum Mechanics

P. Ehrenfest and J. R. Oppenheimer, "Note on the Statistics of Nuclei," Phys. Rev. 37 (1931) 333, http://link.aps.org/doi/10.1103/PhysRev.37.333 , DOI: 10.1103/PhysRev.37.333

Fujita, Ito, and Godoy, Quantum Theory of Conducting Matter: Superconductivity, 2009

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@MichaelBrown: Don't worry, I predict that Congress will pass another copyright extension bill before 2026. – Ben Crowell Aug 27 '13 at 5:34
More to the point: Congress will pass another copyright extension bill before the copyright on Mickey Mouse expires. Who cares about a scholarly paper by a couple of obscure physicists? ;) – Michael Brown Aug 27 '13 at 5:36
Sorry, Ben, I'm unsure exactly what your question is. Are you simply asking about the contents of the Ehrenfest & Oppenheimer paper? Or are you after what modern authors interpret the result to be? – Emilio Pisanty Aug 27 '13 at 15:38
Lipkin's derivation is not relativistic. – akhmeteli Aug 31 '13 at 6:52

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.

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