Elements of Modern Physics
206
Without any loss of generality, we assume that the atom is originally in the
l = 1, m l = 0 state. Using the wave functions given in Sec. 4.1,
| (r) 1s, 2p | = |
1
3 1/2
5 1/2
1
1
exp ( / )
cos
(
)
(32
)
r a
r
z
a
a
−
θ
π
π
∫
exp (– r/2a 1 ) r
2
dr 2πd cos θ |
=
15/2
1
1
5
2
0.745
3
a
a
≈
(6.121)
Substituting this in Eq. (6.70) and using the relation in Eq. (6.77) the lifetime
τ of the 2p state (∆E = ω = 12.0 eV) is
τ ≈ 1.6 × 10
–9
s
(6.122)
which is in agreement with the experimental observation.
Example 5
The ratio of spontaneous transitions to stimulated transitions for particles in
thermal equilibrium can be obtained from Eq. (6.63). Denoting the probability
for spontaneous transitions by P 1 and that for stimulated transition by P 2 ,
Eq. (6.63) reads
P 2 + P 1 = exp ( ω nm /kT) P 2
(6.123)
or
P 1 /P 2 = exp ( ω nm /kT) – 1
(6.124)
For very low temperatures, the transitions are predominantly spontaneous
but become predominantly stimulated for high temperatures. This is to be
expected since the radiation density increases with temperature. For example,
at room temperatures, the transitions between 2p and 2s states of the hydrogen
atom ( ω nm ~ 10 eV, kT ~ 0.026 eV) are predominantly spontaneous. However,
in some of the hot stars, the surface temperatures are as high as 30000 K, so that
there stimulated transitions also are important.
Example 6
Lasers provide an intense, collimated beam. To estimate the power, consider
the original ruby laser which had a diameter of 1 cm and a length of 5 cm. If the
ruby has about 10
19
Cr atoms/cc, and all of them are excited, the total energy
available is
E = 10
19
5
4
π
× (hv)
= 11.25 J
(6.125)
If the pulse lasts for about 10
–7
s, the power during this period is about
10
8
W.
206
Without any loss of generality, we assume that the atom is originally in the
l = 1, m l = 0 state. Using the wave functions given in Sec. 4.1,
| (r) 1s, 2p | = |
1
3 1/2
5 1/2
1
1
exp ( / )
cos
(
)
(32
)
r a
r
z
a
a
−
θ
π
π
∫
exp (– r/2a 1 ) r
2
dr 2πd cos θ |
=
15/2
1
1
5
2
0.745
3
a
a
≈
(6.121)
Substituting this in Eq. (6.70) and using the relation in Eq. (6.77) the lifetime
τ of the 2p state (∆E = ω = 12.0 eV) is
τ ≈ 1.6 × 10
–9
s
(6.122)
which is in agreement with the experimental observation.
Example 5
The ratio of spontaneous transitions to stimulated transitions for particles in
thermal equilibrium can be obtained from Eq. (6.63). Denoting the probability
for spontaneous transitions by P 1 and that for stimulated transition by P 2 ,
Eq. (6.63) reads
P 2 + P 1 = exp ( ω nm /kT) P 2
(6.123)
or
P 1 /P 2 = exp ( ω nm /kT) – 1
(6.124)
For very low temperatures, the transitions are predominantly spontaneous
but become predominantly stimulated for high temperatures. This is to be
expected since the radiation density increases with temperature. For example,
at room temperatures, the transitions between 2p and 2s states of the hydrogen
atom ( ω nm ~ 10 eV, kT ~ 0.026 eV) are predominantly spontaneous. However,
in some of the hot stars, the surface temperatures are as high as 30000 K, so that
there stimulated transitions also are important.
Example 6
Lasers provide an intense, collimated beam. To estimate the power, consider
the original ruby laser which had a diameter of 1 cm and a length of 5 cm. If the
ruby has about 10
19
Cr atoms/cc, and all of them are excited, the total energy
available is
E = 10
19
5
4
π
× (hv)
= 11.25 J
(6.125)
If the pulse lasts for about 10
–7
s, the power during this period is about
10
8
W.