Zdunkowski W., Trautmann T., Bott A. Radiation in the atmosphere: A course in Theoretical Meteorology
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8.3 Some basic principles from quantum mechanics 295
the gradient of L assumes the form
∇L = ev × (∇×A) + ev ·∇A − e∇φ (8.72)
Application of the Euler expansion
dA
dt
=
∂A
∂t
+ v ·∇A (8.73)
yields
d
dt
(mv) = F =−e
∂A
∂t
− e∇φ + ev × (∇×A) (8.74)
in accordance with (8.67)
In Appendix 8.8.1 it will be shown that the Hamiltonian function can be written as
H =
n
k=1
p
k
˙
q
k
− L = p · v − L (8.75)
Since the momentum of the charged particle is given by
p = mv + eA (8.76)
the classical expression for H assumes the form
H = p · v − L =
1
2
mv
2
+ eφ (8.77)
Using (8.76) H can also be written as
H =
1
2m
(
p − eA
)
2
+ eφ (8.78)
For an electromagnetic wave such as that associated with a light wave the
Maxwell conditions
∇·A = 0, φ = 0 (8.79)
apply, so that the Hamilton function will simplify. As shown in textbooks on quan-
tum mechanics the vectors p and A, in general, do not commute but follow the
rule
p · A − A · p = i¯h∇·A (8.80)
Thus, by using (8.79), the Hamilton operator can be written as
H =
p
2
2m
−
e
m
A · p +
e
2
2m
A
2
(8.81)
296 Absorption by gases
Since the perturbation of an atomic system by a light wave will be small, in
discussing radiation we may neglect the last term on the right-hand side of (8.81).
This term, however, cannot be neglected when discussing the perturbations due
to strong magnetic fields. The first term in (8.81) refers to the Hamiltonian of the
particle in the absence of a radiation field while the second term accounts for the
perturbation of the system due to an electromagnetic field. The perturbation part
H
of the operator H is expressed by
H
=−
e
m
A · p =−
e ¯h
mi
A ·∇
(8.82)
where we have also replaced the linear momentum by the corresponding quan-
tum mechanical operator as defined in Table 8.1. The perturbation part H
of the
Hamiltonian is also called the interaction Hamiltonian.
8.3.4 The interaction Hamiltonian
Now we consider a molecular system subjected to the perturbation H
of an elec-
tromagnetic field of a light wave. Since the molecular dimensions are much smaller
than the wavelength of the infrared light, we may consider A to be a constant over
the molecule. For simplicity, we will assume that the field is that of a plane polar-
ized light wave traveling in the z-direction. With E
x
= E
0
exp
[
i(ωt − kz)
]
and
E
y
= E
z
= 0 we find from (8.66)
A
x
=−
E
x
iω
, A
y
= 0, A
z
= 0 (8.83)
which enables us to obtain the interaction Hamiltonian.
By using in (8.82) the expression (8.76) of the momentum p we would again
obtain an expression that is proportional to A
2
which we neglect in first-order
perturbation theory. Thus we simply write for the x-component of the momen-
tum p
x
= m
˙
x. For a forced harmonic oscillation of frequency ω, p
x
also varies
sinusoidally as stated in
p
x
= m
dx
dt
= m
d
dt
[x
0
exp(iωt)] = mi ωx (8.84)
so that the perturbation Hamiltonian is given by
H
=−
e
m
A · p = exE
x
= ex E
0,x
cos ωt
=
1
2
exE
0,x
[exp(iωt) + exp(−iωt)]
(8.85)
8.3 Some basic principles from quantum mechanics 297
The quantity ex E
x
is the product of the dipole moment and the electric field in
the x-direction. Similarly we may consider the remaining directions. If the dipole
moment of the system is denoted by M whose components are
M
x
=
k
e
k
x
k
, M
y
=
k
e
k
y
k
, M
z
=
k
e
k
z
k
(8.86)
where the suffix k refers to particle k, then the interaction energy can be expressed
by M · E.
Any transition for which the probability can be calculated using the form (8.85)
is called an electric dipole moment transition. The effect of the magnetic field has
been ignored. The discussion we have carried out so far is a mixture of classical
and quantum mechanics which is known as a semi-classical treatment.
8.3.5 Computation of transition probabilities
We begin our discussion by restating the wave equation in the form
H = i¯h
∂
∂t
(8.87)
where is the complete wave function. The Hamiltonian operator may be expressed
as H = H
0
+ H
. The part H
0
is independent of time while H
is a time-dependent
perturbation. The unperturbed eigenfunctions
0
satisfy
H
0
0
= i¯h
∂
0
∂t
(8.88)
In order to obtain a solution to (8.87) we expand the function in terms of the
unperturbed eigenfunctions
0
permitting the expansion coefficients to vary with
time. Substituting
=
n
a
n
(t)
0
n
with
0
n
= ψ
0
n
exp
−
iE
n
t
¯h
(8.89)
into (8.87) we find the following equation
n
a
n
(t)H
0
0
n
+
n
a
n
(t)H
0
n
= i¯h
n
da
n
dt
0
n
+ i¯h
n
a
n
(t)
∂
0
n
∂t
(8.90)
Since the unperturbed eigenfunctions satisfy (8.88), equation (8.90) immediately
reduces to
n
a
n
(t)H
0
n
= i¯h
n
da
n
dt
0
n
(8.91)
298 Absorption by gases
Now we multiply both sides of equation (8.91) by
0∗
m
and then integrate over
the coordinate space yielding
n
a
n
(t)
0∗
m
H
0
n
dτ = i¯h
n
da
n
dt
0∗
m
0
n
dτ = i¯h
da
m
dt
(8.92)
Hence, due to the orthogonality of the wave functions we immediately obtain the
result
2
da
m
dt
=−
i
¯h
n
a
n
(t)
0∗
m
H
0
n
dτ =−
i
¯h
n
a
n
(t)
0∗
m
|H
|
0
n
(8.93)
Often the integral is written in the operator form as shown in the final expression
of this equation. For any particular problem we have to solve a set of differential
equations to get explicit expressions for the a
m
.
Temporarily we consider the perturbation for a single frequency. We assume
the simple situation that the system originally at t = 0 was in the state n so that
a
n
(0) = 1 and all the other a
k
are zero at time t = 0. For a sufficiently short
time so that all a
k
are negligible except a
n
, we find the following approximate
expression
a
m
(t) =−
i
¯h
t
0
0∗
m
H
0
n
dτ dt
with a
m
(0) = 0 (8.94)
Making use of
0
n
= ψ
0
n
exp(−iE
n
t/¯h) where the ψ
0
n
depend on space only, and
replacing the perturbation Hamiltonian by (8.85), upon integration we obtain
a
m
(t) =
1
2
E
0,x
X
nm
1 − exp
it
¯h
(E
m
− E
n
+ hν)
E
m
− E
n
+ hν
+
1 − exp
it
¯h
(E
m
− E
n
− hν)
E
m
− E
n
− hν
(8.95)
The expression
X
nm
=
ψ
0∗
m
M
x
ψ
0
n
dτ (8.96)
is called the x-component of the matrix element of the dipole moment for the
transition n to m.
Let us consider the case E
m
> E
n
so that the transition corresponds to absorption.
The coefficient a
m
will be large only if E
m
− E
n
is approximately equal to hν so
that the denominator of the second term on the right-hand side of (8.95) is nearly
2
Even shorter is Dirac’s notation which is written as m|H
|n where the terms m|=
0∗
m
and |n=
0
n
are
known as (bra) and (ket), respectively. The integration over all space is implied.
8.3 Some basic principles from quantum mechanics 299
zero. The first term can be ignored so that the product a
m
a
∗
m
is given with excellent
approximation by
a
m
(t)a
∗
m
(t) =
E
2
0,x
|X
nm
|
2
sin
2
πt
h
(E
m
− E
n
− hν)
(E
m
− E
n
− hν)
2
(8.97)
In case of emission E
n
> E
m
the first expression will be the dominant term and a
similar expression will be found for a
m
a
∗
m
.
Inspection of (8.97) shows that for small t the transition probability varies accord-
ing to t
2
which is an unexpected result. This difficulty is due to the fact that so far we
have considered only a single frequency. From experience we know that we never
deal with strictly monochromatic radiation but always with a range of frequencies
and with radiation fields having components in all three directions.
The energy density of the radiation field in the frequency interval ν to ν + dν
will be denoted by
ˆ
u
ν
dν. From electromagnetic theory it is known that the energy
density and the electric field are related by
ˆ
u
ν
= ε
0
E
2
(ν) (8.98)
where the overbar represents an average value. For isotropic radiation the following
equation is valid
1
3
E
2
(ν) = E
2
x
(ν) = E
2
y
(ν) = E
2
z
(ν) (8.99)
with
E
2
x
(ν) = E
2
0,x
(ν) cos
2
2πνt =
1
2
E
0,x
(ν)
2
(8.100)
Hence the energy density can be expressed by the following relation
ˆ
u
ν
=
3
2
ε
0
E
2
0,x
(8.101)
Assuming that
ˆ
u
ν
is constant over the frequency range ν to ν + dν we may
integrate (8.97) yielding
a
m
(t)a
∗
m
(t) =
2
3ε
0
|X
nm
|
2
ˆ
u
ν
∞
−∞
sin
2
πt
h
(E
m
− E
n
− hν)
(E
m
− E
n
− hν)
2
dν (8.102)
At first glance it seems to be inconsistent to treat
ˆ
u
ν
as a constant and then
integrate over the complete frequency range from −∞ to +∞. Nevertheless, the
approximation is entirely satisfactory since a
m
a
∗
m
is very small except for such
frequencies that E
m
− E
n
= hν. By observing that
∞
−∞
sin
2
α/α
2
dα = π we may
300 Absorption by gases
carry out the integration and find the result
a
m
(t)a
∗
m
(t) =
2π
2
3ε
0
h
2
|X
nm
|
2
ˆ
u
ν
t (8.103)
This relation shows that now a
m
a
∗
m
varies linearly with time t as should be expected.
Corresponding expressions can be obtained for the other directions.
Thus we find for the total transition probability per unit time the expression
a
m
(t)a
∗
m
(t)
t
=
2π
2
3ε
0
h
2
|R
nm
|
2
ˆ
u
ν
(8.104)
with
|R
nm
|
2
=|X
nm
|
2
+|Y
nm
|
2
+|Z
nm
|
2
=
ψ
0∗
m
Mψ
0
n
dτ
2
(8.105)
The term R
nm
is the total matrix element for the transition n to m.
So far our discussion includes only transitions between the so-called non-
degenerate energy levels. Frequently, however, there are several different orthogonal
eigenfunctions associated with one and the same eigenvalue so that we are deal-
ing with a degenerate state. The frequency of this occurrence, i.e. the number of
eigenfunctions corresponding to this state, is the so-called statistical weight.
Suppose that the levels n and m are degenerate with statistical weights g
n
and
g
m
. The probability of transition per unit time from one of its lower states n
i
to the
upper level is given by
a
m
(t)a
∗
m
(t)
t
=
2π
2
3ε
0
h
2
ˆ
u
ν
k
|R(n
i
m
k
)|
2
(8.106)
where the summation must be carried out over all states belonging to the upper
level. Let N
n
represent the number populating the lower level n. Assuming that the
lower states will be equally distributed between the g
n
states, then each state will
have a population N
n
/g
n
. Now the transition probability per unit time is given by
a
m
(t)a
∗
m
(t)
t
=
2π
2
3ε
0
h
2
g
n
ˆ
u
ν
i
k
|R(n
i
m
k
)|
2
(8.107)
8.3.6 Einstein transition probabilities
So far we have discussed the process of absorption and emission in the presence of
an electromagnetic field. Since a system in an excited state can emit radiation even
in the absence of an electromagnetic field, the completion of the theory of radiation
requires the calculation of the transition probability of spontaneous emission. The
direct quantum-mechanical calculation of this quantity is a matter of great difficulty.
8.3 Some basic principles from quantum mechanics 301
Fortunately, Einstein has shown how to tackle the problem of spontaneous emission
by using thermodynamic reasoning.
We investigate the equilibrium between two states of different energy. As stated
in equation (8.46) the transition between two states is always accompanied by the
absorption or emission of radiation. Let us now consider the radiation density in
an enclosure having opaque walls of uniform temperature T containing a large
number of quantized systems which can interact with the radiation. Two states m
and n of these systems have the respective energies E
m
and E
n
. Any transition
between these states will be accompanied by absorption or emission of radiation.
For a wave, considering absorption with E
m
> E
n
,wehave
hν
mn
= E
m
− E
n
(8.108)
We will denote the energy density in the spectral interval ranging from ν
mn
to
ν
mn
+ dν
mn
by
ˆ
u(ν
mn
)dν
mn
. For ease of identification we will momentarily call l
the lower and u the upper energy levels. The probability p
abs
that a system in state
l absorbing a quantum of radiative energy and undergoing a transition to state u in
unit time is given by
p
abs
(l → u) = B
l→u
ˆ
u(ν
lu
) (8.109)
where the coefficient B
l→u
is known as the Einstein coefficient of absorption. The
transition in the opposite direction is given by
p
em
(u → l) = A
u→l
+ B
u→l
ˆ
u(ν
lu
) (8.110)
where the coefficient B
u→l
is the Einstein coefficient of induced emission which is
stimulated emission in the presence of the radiation field of volume density
ˆ
u(ν). In
addition spontaneous emission is taking place which is described by the Einstein
coefficient of spontaneous emission A
u→l
.
3
Within the enclosure the systems will have various energy states. We denote the
number of systems of energy E
l
by N
l
, and the number of systems E
u
by N
u
, then
in equilibrium the number of transitions from u → l must be equal to the number
of transitions from l → u. Therefore, we must have the equilibrium statement
N
l
B
l→u
ˆ
u(ν
lu
) = N
u
[
A
u→l
+ B
u→l
ˆ
u(ν
lu
)
]
(8.111)
Since N
l
, N
u
are numbers per unit volume, each term expresses the number of
transitions in unit time per unit volume. Equation (8.111) can be solved to give the
3
The quantities
ˆ
u(ν
lu
) and B
l→u
are respectively expressed in J s m
−3
and m
3
J
−1
s
−2
while A
u→l
is expressed
in s
−1
.
302 Absorption by gases
ratio
N
u
N
l
=
B
l→u
ˆ
u(ν
lu
)
A
u→l
+ B
u→l
ˆ
u(ν
lu
)
(8.112)
In an enclosure at temperature T we may also express the ratio (8.112) with the
help of the Boltzmann distribution. The number of systems having an energy E
i
above the ground state is given by
N
i
=
N
Z
g
i
exp
−
E
i
kT
(8.113)
where N is the total number of systems and Z the partition function, that is the sum
over all states. The quantity g
i
is the statistical weight of the level of energy E
i
.
Therefore, at equilibrium the following expression is valid
N
u
N
l
=
g
u
g
l
exp
−
E
u
− E
l
kT
=
g
u
g
l
exp
−
hν
lu
kT
(8.114)
Substituting (8.114) into (8.112) and solving for the energy density
ˆ
u(ν
lu
), we obtain
ˆ
u(ν
lu
) =
g
u
A
u→l
g
l
B
l→u
exp
(
hν
lu
/kT
)
− g
u
B
u→l
(8.115)
Within the enclosure the energy density must also be given by Planck’s radiation
law
ˆ
u(ν
lu
) =
8πhν
3
lu
c
3
1
exp
(
hν
lu
/kT
)
− 1
(8.116)
Comparison of (8.115) and (8.116) gives the important relations
A
u→l
=
8πhν
3
lu
c
3
B
u→l
, g
u
B
u→l
= g
l
B
l→u
(8.117)
From time-dependent perturbation theory we obtained equation (8.107) which
expresses the Einstein probability for absorption. Equation (8.109) also states this
probability, but the Einstein coefficient remained undefined. We observe that in
(8.107) the level m refers to the upper level. By setting (8.107) equal to (8.109),
replacing n by l we find the important equation for the Einstein coefficient for
absorption
B
l→u
=
2π
2
3ε
0
h
2
g
l
i
k
|R(l
i
u
k
)|
2
(8.118)
This formula will be needed when we derive the line intensity equation for the
spectral absorption coefficient. Using (8.117) and (8.118) yields the coefficient for
8.3 Some basic principles from quantum mechanics 303
spontaneous emission
A
u→l
=
16π
3
ν
3
lu
3hε
0
c
3
g
u
i
k
|R(l
i
u
k
)|
2
(8.119)
To the degree of approximation we have used above, the Einstein coefficients depend
mainly on the matrix element for the electric dipole moment between the two states.
If the variation of the field over the molecule is not neglected additional terms will
appear. The first two of these correspond to magnetic dipole and electric quadrupole
radiation. In comparison to the transition probability of electric dipole radiation the
contributions of these additional terms may be ignored in most cases. Often these
terms are loosely called forbidden transitions.
8.3.7 Line intensities
We start the discussion with Beer’s law as given by
dI
ν
=−k
ν
I
ν
du (8.120)
Here, k
ν
is the monochromatic absorption coefficient and du the differential absorb-
ing mass. As we know, a spectral line is not infinitely sharp but it is broadenend.
Over the small frequency interval occupied by a spectral line, the radiative energy
from a nearly parallel beam varies so little that I
ν
may be treated as a constant.
Thus we may perform the frequency integration and obtain
dI
ν
=−I
ν
du
∞
−∞
k
ν
dν =−SI
ν
du (8.121)
where we have used the definition (7.13) for the line intensity S. To find a theoretical
expression for S we need to relate this quantity to the net number of transitions
N
tr
from the lower energy level E
l
to the upper energy level E
u
. The number N
tr
induced by a radiation field of energy density per unit volume
ˆ
u(ν
lu
)isgivenby
N
tr
=
(
N
l
B
l→u
− N
u
B
u→l
)
ˆ
u(ν
lu
) (8.122)
Due to (8.117) we may rewrite this expression as
N
tr
= B
l→u
N
l
− N
u
g
l
g
u
ˆ
u(ν
lu
) (8.123)
The energy absorbed in each transition is hν
lu
so that the decrease dI
ν
in the
beam may be expressed as
dI
ν
=−
hν
lu
c
B
l→u
N
l
− N
u
g
l
g
u
I
ν
lu
du (8.124)
304 Absorption by gases
Here we have substituted I
ν
/c for
ˆ
u
ν
. Comparing the latter equation with (8.121)
we find
S =
hν
lu
c
B
l→u
N
l
− N
u
g
l
g
u
(8.125)
With the help of (8.118) we now replace B
l→u
by B
m→n
and obtain
S =
2π
2
ν
mn
3hε
0
c
i
k
|R(m
i
n
k
)|
2
N
m
g
m
−
N
n
g
n
with R(m
i
n
k
) =
ψ
0∗
(m
i
)Mψ
0
(n
k
) dτ
(8.126)
The indices i and k number the degenerate wave functions belonging to the energy
levels m and n, respectively. The summation must be carried out over all possible
combinations of wave functions of the upper state with wave functions of the lower
state.
In case of thermal equilibrium, using the Boltzmann distribution (8.113), the line
intensity can also be expressed as
S =
2π
2
ν
mn
3hε
0
c
i
k
|R(m
i
n
k
)|
2
N
Z
1 − exp
−
hν
nm
kT
exp
−
E
m
kT
(8.127)
The frequency ν
nm
has previously been called ν
0
which refers to the frequency
defining the position of the center of the absorption line. The above line intensity
formulas ignore the effect of nuclear spin which is responsible for the existence of
the so-called hyperfine structure of the spectrum, see Rothman et al. (1987, 1992).
Very few problems have exact quantum mechanical solutions. To this class of
problems belong the harmonic oscillator and the rigid rotator for which exact solu-
tions can be given. More complicated problems require approximate solutions.
8.4 Vibrations and rotations of molecules
8.4.1 The harmonic oscillator
Harmonic oscillation is of considerable importance in quantum mechanics. The
model of the simple harmonic oscillator is used to understand the vibrations of
diatomic and polyatomic molecules. A harmonic oscillator is a particle of mass m
moving in a straight line (say along the x-axis) subject to the potential V = kx
2
/2
where k is Hooke’s constant. According to (8.58) the classical Hamiltonian of the