SPIN, A FUNDAMENTAL PROPERTY OF THE ELECTRON

As described in Chapter 2, Pauli postulated the existence of spin to help Bohr explain the properties of the elements and the periodic table. It worked, but there was no firm scientific basis. That basis was put in place in one magnificent theoretical work by the mathematician and physicist Paul Dirac in 1928.¹ Using Einstein's special relativity and only the electron's electrical charge and mass, he calculated all of the rest of the properties of the isolated electron (that is, the intrinsic properties of the electron, regardless of whether or not it is in an atom).² (Remember that it was Dirac, along with Heisenberg and Schrödinger, each using a different mathematical approach to quantum mechanics, who three years earlier had all successfully calculated the energies and spatial states for the hydrogen atom, and all got the same results, though Schrödinger's format was more amenable to interpretation.)

The properties that Dirac derived included a very tiny intrinsic angular momentum that would be present even if the electron were completely isolated from the atom and regardless of the overall shape of the spatial state that the electron might take on. Even though there is nothing in what Dirac did to indicate that the electron is in any way spinning, the term used to describe this intrinsic angular momentum was (and still is) spin (as coined earlier by Uhlenbeck and Gaudsmit). Dirac calculated that the spin of an electron can have only two values, either +½ or –½ times one basic unit of angular momentum, which, as described earlier, is Planck's constant divided by 2π, written h/2π.³ And so we see that spin is also quantized, with just two possible values, that is, two possible spin quantum numbers. We refer to these here simply as “plus spin” and “minus spin.” (It is because of this binary spin state, that experiments with the photon [with its binary angular-momentum polarization states, as described in Chapter 11] can be straightforwardly substituted for spin in experiments, as was done in the confirmation of Bell's inequality, as described in Chapter 6.)

MAGNETISM

If a bar magnet is placed in a magnetic field (for example, between the jaws of a horseshoe magnet) it will be attracted into the strongest part of the field, and it will line up to point toward a magnetic field's north pole, just as a compass needle points toward the earth's north pole. We say that the bar magnet has a magnetic moment.

If an isolated electron is placed in a magnetic field, it behaves in a similar way to the bar magnet, demonstrating that it too has a magnetic moment. Because the electron always has this magnetic moment regardless of where it happens to be, we say that it has an intrinsic magnetic moment. And quantum mechanics shows that the electron's intrinsic magnetic moment has a magnitude and direction in proportion to its spin angular momentum.⁴ If the electron is in a plus spin state, then its magnetic moment aligns with the magnetic field as described, and if it is in a minus spin state, it aligns in the opposite direction.

Remember from Chapter 11 that the component of the spatial-state angular momentum that will line up in the direction of a magnetic field is quantized as a part of each spatial-state solution to Schrödinger's equation. This alignment occurs because the electron also has a spatial-state magnetic moment in proportion to this same component of spatial-state angular momentum. And remember that this component is quantized and represented by an integer that we refer to generally as m. (So we now understand why m is referred to as the “spatial-state magnetic quantum number.”)

It is the combination of intrinsic spin and m that determines the magnetic properties of the electron for each combined spin and spatial state that it may be in (occupy). There is a very small quantized shift in the energy for each spin and spatial-state combination (thinking classically for the moment), depending on how the tiny “bar magnets” of the electron's intrinsic magnetic moment and spatial-state magnetic moment line up with each other to attract or repel. This “fine-structure splitting” is so small that we can ignore it here, but each energy shift has been measured and verified to occur exactly according to theory.