QUANTUM THEORY 
61 
lúe from 
quantum 
Allowing 
es q 2 and 
ntisation 
le effects 
model is 
uth inde- 
;e of the 
•efore the 
o another 
epend on 
3 ns occur 
)h results 
;y correc- 
the apse- 
mtisation 
ie system 
us betray 
•gy corre- 
magnetic 
ntroduces 
rovides a 
3orrection 
ires of the 
tion. The 
bservable 
In actual 
he energy 
r alue, and 
small but 
tisation is 
esponding 
the sub- 
se we can 
principal 
1 of reson- 
mmber of 
cycles described without interruption; or the revolutions of the coordinate 
correspond to the lines of a grating (in time dimension) which has higher 
resolving power in proportion to the number of waves that it superposes. 
Orbits of Large Quantum Number. 
44. Material at high temperature contains in addition to the electrons 
bound to the atoms a number of free electrons broken loose from the atoms 
and moving as independent molecules. Statistics of the bound electrons 
are naturally given in the form—number with orbits of such and such 
quantum specification. Statistics of free electrons are given in the form 
—number within given limits of position and velocity. Now there is an 
important continuity between the statistics of bound and free electrons 
which is hidden when they are classified on different principles. Our 
purpose in this section is to transform the statistics of the most loosely 
bound electrons so as to make them comparable with those of the free 
electrons. 
We have seen that the numbers of systems in two states with 
energies Xx > X 2 are iu the ratio 
qi e-x'l RT : q 2 e~x*l RT , 
where the weights q x , q 2 depend on the states but not on the temperature. 
Hence in any assemblage we may set the number of systems in a state 
with energy Xn equal to 
Bq n e~ Xn l RT (44-1), 
where B is a constant depending on the extent of the assemblage. 
We shall assume that the weight of every quantised orbit is the same. The 
general coherence of this assumption with the ideas of statistical mechanics 
will appear later. The weight of each quantised orbit is taken to be unity, 
thus fixing the unit of q which was previously left undefined. 
Consider as in the last section the system consisting of a nucleus 
attended by a single electron. There are n {n + 1) different orbits with 
principal quantum number n, hence the number of systems with energy 
Xn will be 
Bn (n + 1 ) e~ Xn l RT (44-2), 
where by (42-62) — y n = Kjn 2 , K = 27T 2 me i Z 2 /h 2 (44-3). 
Electrons with very small negative energy correspond to large values 
of n. We shall consider n so large that the series of values of the energy 
fades into a practically continuous range. Then by (44-3) 
, _2K 
^Xn 
so that the number of integral values of n in a range dx n approaches 
nHxJ2K (44-4).