64 
QUANTUM THEORY 
By comparison with (44*1) it follows that the weight to be attributed 
to this range is 
-p dxdydzdudvdw (45-6). 
We have thus connected the weight of states specified by a space and 
velocity distribution of electrons with the weight of states specified by 
quantum orbits. 
Writing in (45-5) Xn — — «A + \m ( u 2 + v 2 + w 2 ), we see that these 
electrons with small negative energy obey Maxwell’s Law just as free 
electrons with positive energy do. 
Ionisation. 
46. Suppose that in the foregoing assemblage there are free electrons 
which in regions of zero potential are distributed with density ct 0 . By 
Maxwell’s Law the number in a range dxdydzdudvdw is* 
(fTl!? 7 ) a o e ~ m{u2 + v2 + W ' 2) / 2KT dxdydzdudvdw, 
and generally at places where the potential is not zero the number is 
3 
°oe-xl RT dxdydzdudvdw (46-1), 
\2iTTJtil J 
where y is the kinetic and potential energy. 
We have now two formulae (45-5) and (46-1) for calculating the dis 
tribution of electrons of zero energy, according as zero energy is considered 
to be the limit of small negative or small positive energy. It is reasonable 
to assume that the two formulae must agree. Hence we have a means of 
determining the constant B in terms of cr 0 . When the constants in the 
two formulae agree, we have complete continuity at zero energy. The 
classical formula (46T) does not at first fail when applied to bound 
electrons subjected to quantum restrictions; only when n becomes small 
is the deviation manifested. This is an example of the Correspondence 
Principle which asserts that as n increases the quantum laws approach 
the classical laws as a limit. 
In formulating this continuity we have to proceed carefully because 
our discussion of bound electrons has been confined to the case in which 
there is only one electron attached to the nucleus. We have therefore to 
consider the continuity between the number of systems consisting of a 
nucleus and a single bound electron in a given volume-element and the 
number of systems consisting of an ionised nucleus and a free electron in 
a corresponding volume-element. Let N be the number of nuclei with 
not more than 1 bound electron and Nx the number (out of these) with 
* The constant (m/VwRT)? is found by equating the integral for all values of 
u, v, w to cr 0 dxdydz.