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By D.R. BATES (Eds.)

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18) The left-hand side stands for the rth moment of the statistical aggregate of the measured values, defined in books on statistics as pr=2JwsP: (19) 2. F U N D A M E N T A L P R I N C I P L E S OF Q U A N T U M MECHANICS 43 provided is the relative frequency of the occurrence of the sth eigenvalue pg in the set of measurements. In accordance with (18) the state function predicts not only the mean but all moments of the aggregate of measurements. Now (18) may be transformed as follows. Let the eigenfunctions of Ρ be denoted by φχ, so that Ρφχ = Ρχφχ.

4. Electron Spin In this section we look for the simplest operator (and we will take it to be in matrix form) which will satisfy (35). It will turn out to be the operator discovered by PauH, representing the spin of an electron and many other elementary systems. T o avoid the impression that this is merely a math­ ematical exercise, a few words will first be said concerning the history of the spin problem. 2. F U N D A M E N T A L P R I N C I P L E S OF Q U A N T U M M E C H A N I C S 55 Early quantum theory, in the picturesque form of Bohr's quantum rules as well as in the maturer versions of Heisenberg and Schrödinger, led to some false predictions, particularly with regard to the finer details of the energy states of atoms, the Zeeman effect, and the magnetic properties of electrons.

The question arises, therefore, as to the logical connection between the commutation rules and the form of the operators as given by (33). Commutation rules and operator form might, for instance, be entirely equivalent, in which case (2) would define the forms (33). Or it might be that the commutation rules define a wider class of operators than (33); it could not define a narrower class because we have already seen that (33) is a sufficient condition for (2). As a matter of fact the commutation rules do define a wider class, as we shall show.

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