bei48482_FM

energy density u( ) and also to the properties of states iand j, which we can include

in some constant Bij. Hence the number Ni→jof atoms per second that absorb photons

is given by

Ni→jNiBiju( ) (9.43)

An atom in the upper state jhas a certain probability Ajito spontaneously drop to

state iby emitting a photon of frequency. We also suppose that light of frequency

can somehow interact with an atom in state jto induce its transition to the lower state

i. An energy density of u( ) therefore means a probability for stimulated emission of

Bjiu( ), where Bji, like Bijand Aji, depends on the properties of states iand j. Since Nj

is the number of atoms in state j, the number of atoms per second that fall to the lower

state iis

Nj→iNj[AjiBjiu( )] (9.44)

As discussed in Sec. 4.9, stimulated emission has a classical analog in the behavior

of a harmonic oscillator. Of course, classical physics often does not apply on an atomic

scale, but we have not assumed that stimulated emission doesoccur, only that it may

occur. If we are wrong, we will ultimately find merely that Bji0.

Since the system here is in equilibrium, the number of atoms per second that go

from state ito jmust equal the number that go from jto i. Therefore

Ni→jNj→i

NiBiju( )Nj[Aji Bjiu( )]

Dividing both sides of the latter equation by NjBjiand solving for u( ) gives

u(^ )u(^ )

u( ) (9.45)

Finally we draw on Eq. (9.2) for the numbers of atoms of energies Eiand Ejin a

system of these atoms at the temperature T, which we can write as

NiCeEikT

NjCeEjkT

Hence

e(EiEj)kTe(EjEi)kTehkT (9.46)

Ni



Nj

AjiBji







N

N

i

j







B

B

i

j

j

i





^1

Aji



Bji

Bij



Bji

Ni



Nj

Number of atoms

that emit photons

Number of atoms

that absorb photons

Statistical Mechanics 319

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