116
2 Quantum Mechanics – I
Term
S
P
D
F
l
0
1
2
3
Parity = (−1)
l
+1 −1 +1 −1
2.47 J = l + s = 0 + 1/2 = 1/2
F = I + J, I + J − 1, . . . I − J
= 2, 1, 0
2.48 The observed frequency (ω) of radiation from an atom that moves with the
velocity v at an angle θ to the line of sight is given by
ω = ω 0 (1 + (v/c) cos θ)
( 1 )
where ω 0 is the frequency that the atom radiates in its own frame of referenece.
The Doppler shift is then
Δω
ω 0
=
ω − ω 0
a 0
=
v
c
cos θ
(2)
As the radiating atoms are subject to random thermal motion, a variety of
Doppler shifts will be displayed. In equilibrium the Maxwellian distribution
gives the fraction
dN
N
of atoms with x-component of velocity lying between v x
and v x + dv x
Fig. 2.4 Thermal broadening
due to random thermal
motion
dN
N
=
exp
−
vx
U
2
√
π
dv x
U
(3)
where u/
√
2 is the root-mean-square velocity for particles of mass M at temperature T . Now
u =
2kT
M
1/2
(4)
where k = 1.38 × 10
−23 J/K is Boltzmann’s constant.
Introducing the Doppler widths Δω D and Δλ D in frequency and wavelength
Δω D
ω 0
=
Δλ D
λ 0
=
U
c
=
2kT
Mc 2
1/2
(5)