m to the state n. The — carred out on a constituent dt Leaving matrices on one side for the moment and turning back to wave mechanics, we find that Schroedinger's function in the particular solutions of the wave equation has the property which is an expression of the equivalence of two operations when carried out on the function &. This last equation, or equivalence, is used to define the "differential quotient" of a matrix x with respect to the matrix q, the expressions px and xp being identified in an appropriate way with the two products of the matrices p and x. With the help of the Hamiltonian equations (59) we easily find that Briefly, we may say that matrix mechanics simulates Hamil tonian dynamics, but instead of numbers we have matrices or "q" numbers, as they are sometimes called, and instead of ordinary differentiation we have the processes defined by (65) (66) and (67).
If a matrix y in the new mechanics is "constant," i.e., if 1 it can be shown that it is a diagonal matrix. The constituents y,„„ for which m is different from n are all zero. The matrix H itself must therefore be such a diagonal matrix. Now the rule for multiplying matrices (or determinants) makes the mn consti tuent of the product AB of two matrices A and B equal to the Sum This is, of course, Bohr's postulate, and we learn that the consti tuents of the diagonal matrix H are identical with the energy levels of the older theory.
It will be observed that if we replace x in equation (65) by q we have where ------ expresses equivalence. We shall not discuss the sig
nificance of the ambiguity of sign beyond saying that it cancels out in virtue of the corresponding ambiguity in the product of two matrices. We have further where the right hand side means a matrix (the unit matrix) the diagonal constituents of which are all equal to unity and the others zero. Equation (69) occupies a position in matrix me chanics corresponding to that occupied in the older theory by the Wilson-Sommerfeld conditions (4i ).
Much has been written in recent times about the irrational character of the quantum theory. This irrationality is simply an expression of the difficulty—perhaps of the impossibility—of a coordination of quantum phenomena in the old-fashioned causal space-time manner. It seems possible to retain the notion of elementary particles, electrons, photons, etc., located in space and time—or more probably in a 5-dimensional continuum. If we do this it would appear that the de Broglie-Schroedinger un dulations become merely a mathematical implement for com puting probabilities, and cannot be regarded as physical entities in the ordinary sense of that term. There appears to be no determination in small scale events, except such of a statistical kind, and the very sharply-defined extrinsic causality of the macroscopic world has its being in the fact that probabilities may be so great as to be practical certainties.