When Dirac succeeded in writing a covariant quantum equation for the electron in 1927, he and the physics community got some big shocks. First, the equation automatically contained electron spin. It could not be left out. Second, the equation had a second solution, describing the antiparticle to the electron. The implication was that every particle in nature has a corresponding antiparticle, required by combining quantum physics and relativity. Third, when applied to the hydrogen atom, Dirac's equation automatically contained a potential energy term that looked like V_so(r)L·S, a so-called “spin-orbit” interaction. The result was to shift the hydrogen energy levels around, higher or lower, by roughly 10^-5 eV. This is called “fine structure.”

There is also a term taking into account the interaction of the magnetic moment of the proton with the magnetic moment of the electron. This produces a so-called “hyperfine structure,” which we will not further discuss. It typically changes energies also by about 10^-5 eV. Finally, the fact that the electromagnetic field is quantized causes the central potential energy not to depend on a specific r, but rather a narrow range of possible values. This results in a famous effect called the Lamb Shift, which causes an energy change of about 10^-6 eV in s states. There are also other tiny effects due to the quantization of the field. We will not discuss anything but the fine structure due to the spin-orbit term in the interaction.  To include the spin-orbit effect we introduce a vector operator J that satisfies J = L + S. The result is J² = j(j+1)ℏ² with j = ℓ ± 1/2. J_z = m_jℏ with m_j ranging from -j to j in steps of 1. The resulting usual label for states is of the form nX_j. For example: 2P_3/2 and 2P_1/2.  The usual set of quantum numbers required to specify a state would then be something like [n, ℓ, j, m_j, m_s].

No two identical fermions can go into the same state in the same region of space. Why not? A fundamental principle of quantum physics is that when two or more possibilities cannot be distinguished, the corresponding state functions must be added, creating the possibility of destructive interference between possibilities!

Suppose we have two particles, 1 and 2, and two possible states for such particles, α and β. Quantum physics assigns an amplitude, a, for every possible state and process in nature. If we put particle 1 into state α and particle 2 into state β, the corresponding amplitude is a_α(1)a_β(2).

But there is a basic concept of quantum physics: when two or more situations are completely indistinguishable, the various possibilities combine algebraically. And there is another basic concept: for fermions, the combinations have minus signs, while for bosons they have plus signs. Thus if particles 1 and 2 are absolutely identical fermions, the actual amplitude for the situation in which we are interested is

Since there is no distinction between particles 1 and 2, the two possibilities cannot be distinguished. And this is why “solid” matter appears to be solid! The atoms in the chair seat have electrons in the same states as atoms in you. If the atoms in the chair seat could penetrate into the atoms of you, even slightly, there would be regions of space where two identical electrons are in the same state. What is the probability that this can happen, that two electrons can both be in state α, for example?