Given an arbitrary direction z (usually determined by an external magnetic field) the spin z-projection is given by
where ms is the secondary spin quantum number, ranging from −s to +s in steps of one. This generates 2s+1 different values of ms.
The allowed values for s are non-negative integers or half-integers. Fermions (such as the electron, proton or neutron) have half-integer values, whereas bosons (e.g. photon, mesons) have integer spin values.
Next, the eigenvectors of and satisfy:
The solutions required each possible state of the electron to be described by three "quantum numbers", n, l, and m. These were identified as, respectively, the electron "shell" number, n, the "orbital" number, l, and the "orbital angular momentum" number m. Angular momentum is a so-called "classical" concept measuring the momentum of a mass in circular motion about a point. The shell numbers start at 1 and increase indefinitely. Each shell of number n contains n² orbitals. Each orbital is characterized by its number l, where l takes integer values from 0 to n-1, and its angular momentum number m, where m takes integer values from +l to -l. By means of a variety of approximations and extensions, physicists were able to extend their work on hydrogen to more complex atoms containing many electrons.
Atomic spectra measure radiation absorbed or emitted by electrons "jumping" from one "state" to another, where a state is represented by values of n, l, and m. So called "Transition rule" limit what "jumps" are possible. Generally a jump or "transition" is only allowed if all three numbers change in the process. This is because a transition will only be able to cause the emission or absorption of electromagnetic radiation if it involves a change in the electromagnetic dipole of the atom.
However, it was recognized in the early years of quantum mechanics that atomic spectra measured in an external magnetic field (see Zeeman effect) cannot be predicted with just n, l, and m. A solution to this problem was suggested in early 1925 by George Uhlenbeck and Samuel Goudsmit, students of Paul Ehrenfest (who rejected the idea), and independently by Ralph Kronig, one of Landé's assistants. Uhlenbeck, Goudsmit, and Kronig introduced the idea of the self-rotation of the electron, which would naturally give rise to an angular momentum vector in addition to the one associated with orbital motion (quantum numbers l and m).
The spin angular momentum, is characterized by a quantum number; s = 1/2 specifically for electrons. In the an analogous way of other quantized angular momenta, L, it is possible to obtain an expression for the total spin angular momentum:
The hydrogen spectra fine structure is observed as a doublet corresponding to two possibilities for the z-component of the angular momentum, where for any given direction z:
which solution has only two possible z components for the electron. In the electron, the two different spin orientations are sometimes called "spin-up" or "spin-down".
The spin property of an electron would classically give rise to magnetic moment which was a requisite for the fourth quantum number. The electron spin magnetic moment is given by the formula:
and by the equation:
When atoms have even numbers of electrons the spin of each electron in each orbital has opposing orientation in different directions. However, many atoms have an odd number of electrons or an arrangement of electrons in which the number of "spin-up" and "spin-down" orientations are not the same. These atoms or electrons are said to have unpaired spins which are detected in electron spin resonance.
In 1930, Paul Dirac developed a new version of the Schrödinger Wave Equation which was relativistically invariant, and predicted the magnetic moment correctly, and at the same time treated the electron as a point particle. In the Dirac equation all four quantum numbers including the additional quantum number s arose naturally during its solution.