QUANTUM THEORY 
47 
i 
>erature. 
jerature, 
stributed 
i laws of 
y of the 
intaining 
s at least 
removed 
n X rays 
>ry— 
mied into 
, in finite 
xantum is 
universal 
antaneous 
Drption or 
a” in the 
5 is merely 
i quantum 
Xi> X 2 re_ 
y emitting 
rom state 1 
Law II the 
...(36-1), 
i matter in 
any other 
îs of xi an( i 
ir argument 
Deak of the 
y (1) and (2); 
ansformation 
show that no 
number of atoms with energy y instead of the number in a range y to 
X + d X- 
Let n 1} n 2 be the number of atoms in states 1 and 2 and let I (v 12 ) be 
the energy-density of radiation of frequency v 12 . At present we do not 
assume equilibrium. 
Passage from state 1 to 2 with absorption of radiation will be impossible 
unless radiation of the required frequency is present. The number of 
transitions will vanish if I (iq 2 ) vanishes and presumably will increase 
proportionately to I (iq 2 ); it will also be proportional to the number of 
atoms % capable of this transition. We therefore set the number of transi 
tions in time dt equal to 
(^ 12 ^ 1 d (^ 12 ) dt (36*21), 
where a 12 is an atomic constant. 
Passage from state 2 to state 1 with emission of radiation can occur 
spontaneously without the presence of extraneous radiation. The pro 
portion of atoms spontaneously making this jump per unit time must be 
an atomic constant. We therefore set the number equal to 
b 21 n 2 dt (36*22). 
It is conceivable that these passages may be hindered or stimulated by 
the presence of radiation of frequency v 12 . If so, the diminution or addition 
will presumably be proportional to the intensity of the radiation. We 
therefore set the number of additional passages equal to 
®2i^ 2 d (^ 12 ) dt (36*23), 
where a 21 may be positive or negative. 
The constants a 12 , a 21 , b 21 relate to processes in which the atoms act 
individually and do not depend on any statistical properties of the as 
semblage. In particular, they do not depend on the temperature—in fact 
as yet the assemblage is not supposed to have a temperature. 
Apply these results to an assemblage in thermodynamical equilibrium 
at temperature T, the transitions (36*22) and (36*23) must balance (36*21) 
by Law I. The result is Einstein’s equation 
^i 2 n i d { v i 2 > d ) — b 2 ^n 2 + a 21 n 2 I ( v 12 , T ) (36*3), 
where / is no longer arbitrary but represents the distribution law of 
radiation in equilibrium at temperature T. 
This gives 
n i _ a 21 f j _j ^21 
n 2 cii 2 \ n 21 / (v 12 , T) 
(36*4), 
a formula giving the relative proportions of atoms in the two states in 
material at temperature T.