¹ To clarify the meaning of “quantum number” to the non-specialist we exemplify it by reference to two physical instances: the oscillator (a vibrating mass point) and the H-atom. According to the older Bohr theory, the possible paths of the oscillator are characterized by one integer, n, and this integer also appears in the formula for its energy: E = nh v. The quantity h v is usually called a quantum of energy; hence the energy is “quantized” and n is the quantum number.

With respect to the H-atom the situation is a little more complicated. Since the electron moves about the proton in three dimensions, the Bohr theory needed 3 integers to specify a possible orbit. These same three integers appear in the formula for the energy (though no longer in so simple a form) which is therefore again quantized. The three integers are again called quantum numbers.

In present theory, the Bohr orbit is dissolved into a probability distribution in space, but this distribution is still characterized by integers. These have retained the name quantum numbers. They appear in the energy formulas in much the same way as before.

² To be historically accurate, it should be remarked that at the time of Pauli's discovery the spin was not completely understood. Hence Pauli's form of the principle was: An atom can contain only two electrons having any given set of quantum numbers (exclussive of the spin quantum number, which can take on two values).

³ For accuracy's sake we should also say that the inclusion of the “meson,” the most recently discovered of all elementary particles (there may be several species of mesons) may have been hazardous since its properties and functions are not completely known.

⁴ To simplify the discussion we have omitted the fourth, or spin variable. Thus, to make the subsequent remarks literally precise, the reader should regard elementary particles of different spins as belonging to different species. This simplification does no violence to the basic formalism and need not disturb the philosophical reader.

⁵ H. Margenau, Monist, XLII, 161, 1932; Phil. of Science 4, 337, 1937.

⁶ See H. Weyl, Raum, Zeit, Materie; 3rd ed., §20.

⁷ This raises the question: In what sense is a statement specifying what can be observed and what not, a postulate? Most of the basic principles of physics operate on a more abstract plane, they affect the logical tenets from which such specific propositions flow. Taken out of its context, the statement concerning the non-observability of particle exchanges appears indeed like an inductive generalization. Be that as it may, many physicists have no dislike of such logical inversion; witness the tenor of many leading textbooks which endeavor to “derive” the entire formalism of quantum mechanics from observations. The situation can be saved in this way. What later turns into an axiom is first “discovered” inductively. The result is then phrased as a postulate pregnant with more deductive implications than the discovery warranted. If the additional consequences are also verified, the statement is retained in its axiomatic status.

⁸ E. C. Kemble, Fundamental Principles of Quantum Mechanics, McGraw-Hill, p. 336.

⁹ Obviously, we have omitted here (for simplicity) a statement of the accompanying increase in the charge on the nucleus.