XVII. 4] THE NEW QUANTUM MECHANICS 237 
E. It turns out that the only E values for which solutions exist 
that are continuous, finite and single-valued throughout the 
whole space are the following: 
(1) E>o. 
(2) E = — 27i 2 me*/h 2 n 2 {n = I, 2, 3, 4 . . .) . . 17 : 47 
The first set of solutions corresponds to the hyperbolic orbits 
in ordinary mechanics. It is indeed remarkable that in this 
method of treatment these orbits appear quite spontaneously 
from the fact that every positive value of E leads to finite 
solutions. But what is even more remarkable is that the second 
set corresponds exactly to Bohr’s stationary energy levels of 
the circular or elliptic orbits. The selected values of E may be 
named “ characteristic values ” (Eigenwerte), and the solutions 
of the equation for ip belonging to them “ characteristic 
functions.” 
Schrodinger gives an interesting discussion of the question 
why the equation for y should possess finite solutions only for 
certain selected values of the constant E. Briefly it may be stated 
that the first set of E values, E >> o, makes the square of the 
phase-velocity positive throughout the space, the second set 
makes this quantity negative. In the latter case “ there is in the 
course of time a tendency to exaggerate infinitely all ‘ humps ’ 
of the function and even spontaneously to form humps out of 
quite insignificant traces. Evidently a function which is subject 
to such a revolutionary sort of equation, is continually exposed 
to the very highest danger of increasing or decreasing beyond 
all limit. At any rate it is no longer astonishing that special 
conditions must be fulfilled to prevent such an occurrence. 
The mathematical treatment shows that these conditions consist 
exactly in E having one of the second set of characteristic 
values.” 
Another problem which may be solved by the methods of 
undulatory mechanics is that of the harmonic oscillator. This 
yields for the E levels the values (n + \)hv Q (where in this 
case n may take the values 0, 1, 2, 3, 4 . . .) instead of nhv 0 
as in the ordinary quantum theory. There is, then, this addi 
tional term \hv 0 corresponding to what was suggested as being a 
zero-point energy in some of the previous theories. 
Another interesting problem solved by the new method is 
that of a rotating body such as a molecule. If this body have 
moment of inertia I, the E levels are given by n(n + i)h 2 /8jc 2 l. 
In the ordinary theory instead of n(n + 1) we get n 2 . 
In our actual observations we are interested only in differences 
of level, and in this case the new formula amounts to the same 
as replacing n 2 by (n + f) 2 , f° r { n + |) 2 — + 1) —