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E 2 N 2
E 2 − E 1
ω =
ħ
E 1 N 1

Figure 3.1 Two-level atomic system interacting with electromagnetic radiation.

E 2 N 2 E 2 N 2

E 1 N 1 E 1 N 1
(a) (b)

Figure 3.2 Two-level atomic system absorbing a photon (a) Before absorption and (b) After absorption.

outer orbit. To make this transition, atoms require energy corresponding to the difference in energy levels and
this is provided by the incident electromagnetic radiation. The rate of absorption depends on the population
density in the level E and also on the energy spectral density per unit volume of the radiation. Einstein
1
postulated that the number of atoms undergoing absorption per unit time per unit volume from level 1 to
level 2 is given by
( )
dN 1
R abs ≡ − = B u ()N , (3.2)
1
12 s
dt
abs
where u () is the electromagnetic energy spectral density per unit volume, B is a constant, and R is the
s 12 abs
rate of absorption. The negative sign in Eq. (3.2) indicates that the population density in level 1 decreases
3
due to absorption. For example, consider an atomic system of volume 1 m .If10 15 atoms make an upward
3
transition per second after absorbing the incident radiation in a volume of 1 m , the absorption rate is
15 −1
−3
R = 10 s m . This also means that 10 15 photons are absorbed per second per cubic meter.
abs
(b) An atom which is in the excited state of energy E is stimulated to emit radiation at frequency
2
=(E − E )∕ℏ if the radiation at that frequency is already present. After emitting the radiation, it goes
2
1
to state of energy E , as shown in Fig. 3.3. This process is called stimulated emission. Einstein postulated
1
that the rate of emission is proportional to the energy spectral density of radiation at frequency and the
population density at the excited state E ,
2
( )
dN 2
R ≡ − = B u ()N , (3.3)
stim 21 s 2
dt
stim
E 2 N 2 E 2 N 2
ħω
E 1 N 1 E 1 N 1
(a) (b)
Figure 3.3 Two-level atomic system emitting a photon due to stimulated emission (a) Before stimulated emission and
(b) After stimulated emission.