1000 Solved Problems in Modern Physics

(Tina Meador) #1

116 2 Quantum Mechanics – I


Term SPDF
l 0123
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 angleθto the line of sight is given by


ω=ω 0 (1+(v/c) cosθ)(1)

whereω 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

)

cosθ (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 fractiondNNof atoms with x-component of velocity lying betweenvx
andvx+dvx

Fig. 2.4Thermal broadening
due to random thermal
motion


dN
N

=

exp

[


(vx
U

) 2 ]


π

dvx
U

(3)

whereu/


2 is the root-mean-square velocity for particles of massMat tem-
peratureT.Now

u=

(

2 kT
M

) 1 / 2

(4)

wherek= 1. 38 × 10 −^23 J/K is Boltzmann’s constant.
Introducing the Doppler widthsΔωDandΔλDin frequency and wavelength
ΔωD
ω 0

=

ΔλD
λ 0

=

U

c

=

(

2 kT
Mc^2

) 1 / 2

(5)
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