714 X. ON THE STRUCTURE OF 'MATTER'
Section E. The new quantum mechanics.
The main problem of the quantum theory is to determine these functions of the operations, so that the solution of certain equations (hamiltonian) may represent the experimental facts. The original equations of the new mechanics of Heisenberg, Born, and Jordan were frankly founded on an empirical basis. As Dirac puts it, in seeking for the new equations, the classical equations were to be retained as far as possible and only the operations by which these quantities are combined were to be altered.
To gain this freedom to alter multiplication the data were first interpreted as matrices. Then it was found, by Born and Wiener, that the matrices could be interpreted as a special kind of operator, which furnished means to calculate the matrices. Carl Eckart independently developed a simple operator calculus for the solution of the quantum problems. In this present work I follow closely the paper of Eckart.7
The origin of the new quantum mechanics was an epoch-making paper by Werner Heisenberg, in July, 1925. The older quantum theory had postulated the existence of stationary states of the atom, which were calculated with the aid of the older mechanics. In the new mechanics the equations have similar form as in the classical theory, but the variables no longer obey the commutative law of multiplication. In general pq is not equal to
in the classical theory) but, where h represents the Planck con-
stant, q the generalized co-ordinate, p the momentum, 1 stands for the unit matrix, andhave the usual meaning. The fact that multiplication is
not commutative in our calculus allows us to give a definite value to the above difference and by introducing the Planck constant h, we are enabled to introduce the quantum conditions in our calculations.
The quantum conditions of the older theory led to an algebraic equation. By using the classical equation with a non-commutative multiplication law for the variables it is possible to perform calculations in the new and wider scheme of dynamics. The difference between pq and qp is expressed in terms of the Planck constant h. When h is made to approach zero, pq approaches qp, and so we pass to the classical mechanics. Thus we see that the classical mechanics appear only as a particular case of this more general theory.
In introducing his theory, Heisenberg pointed out that the older mechanics uses quantities which are never observable, and can never be observed, such as, for instance, orbital frequencies and amplitudes, or the position and 'time' of revolution of an 'electron'. , which, as such, have no physical meaning. He proposes to use observable data, such as the frequencies and intensities of the radiations , . Now these frequencies are always differences between two terms given by integers. are two such terms, the observable frequency
is theoretically represented by. Such numbers as characterize
the atom as far as it is observable. It was natural that such a collection ('sum' in this case has no longer any physical meaning) of terms could best be represented by a matrix. In the classical theory a dynamical quantity was