QUANTUM THEORY
47
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Law II the
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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.