‘U4,'d£: T_i-Hlibt (1)

'has in general no stationary solution (we put the Planck constant h equal to Zn). But in the limit when the change of H, is made infinitely slow, the system, when started from a stationary state of HO’, passes through the corresponding stationary. states of H, for all t. This is the assertion of the adiabatic theorem of quantum mechanics; The most complete proof hitherto given to the adiabatic theorem is due to Born and Fockfll-(hereafter quoted as BF). Their proof is very general so far as concerns the occurrence of crossing of eigenvalues, but in other respects it is still restricted two essential assumptions: it the spectrum of H, consists of-purely discrete eigenvalues;-ii) these eigenvalues are non-degenerate except for accidental degeneracy caused by crossing.‘Their argument cannot be extended to more general case without radical modification, for they consider simultaneously infinite number of solu-tions of (1) corresponding to the complete system of discrete eigenfunctions; which does not exist in general case.

From thephysical point of view, these as-sumptions i), ii) are rather artificial, for it is not plausible that the solution of (1) corresponding to some particular eigenvalue/1, of H, should be influenced essentially by the nature, discrete or continuous etc., of those parts of the spectrum which are distant from 1,. In the present note we shall give a new