Helium

Having dwelt lovingly upon the Zeeman effect, I will pass more fleetingly over some of the other applications of spin. I wish only to give an inkling of the pervasiveness of the concept.

Close study of the spectrum of helium revealed a striking feature. There are two systems of energy levels, which do not interact with each other-- transitions between the systems do not exist. The bottom level of the first system is well below the bottom level of the second system. I omit a detailed catalog of other regularities.

Born, Heisenberg, Pauli, and other applied more and more recondite techniques from classical mechanics to the problem of the helium spectrum (specifically celestial mechanics, including Poincaré's work). All in vain. As I've already noted, helium poses one of the simplest three-body problems in quantum mechanics, and the old quantum theory just couldn't handle it.

Heisenberg solved the problem after the invention of quantum mechanics. The two electrons each have spin-$\frac{1}{2}$, so the total spin $S$ is either 0 or 1. States with $S=0$ are called singlet states, states with $S=1$ are triplet states (recall again the formula $2S+1$ for the number of possible values for $M_S$). The selection rule $\Delta S = 0$ accounts for the division of energy levels into two subsystems. If $S=1$, then the electrons have the same spin, and cannot both occupy the bottom level in a term scheme like figure 1. If $S=0$, they can. Thus the bottom singlet state is significantly lower than the bottom triplet state. The bottom triplet state is sometimes called metastable, for processes other than photon absorption or emission can (slowly) change a triplet state into a singlet state. (The $\Delta S = 0$ selection rule applies strictly only to electric dipole radiation transitions.)

Heisenberg noted that the hydrogen molecule H$_2$ should also have a similar ``double spectrum'', corresponding to $S=0$ and $S=1$ states. (Here we ignore the spins of the protons and count only the spins of the two electrons.) In a sense, H$_2$ has two ``allotropic forms'', dubbed parahydrogen and orthohydrogen. These two forms were subsequently discovered by experimentalists. The Nobel committee cited this work when awarding Heisenberg the Nobel prize.

© 2001 Michael Weiss

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