Chemistry 342
Spring, 2005
Quantum Mechanics of Light Absorption.
We consider here the quantum mechanics of the absorption of light, between two states 1
and 2, whose eigenfunctions and eigenvalues are ψ1, ψ2 and E1, E2, respectively.
E2-------E1--------hν
ΔE  E 2  E1
c
 hν  h 
λ
Clearly, absorption is “allowed” when the energy of the photon, hν, is equal to the energy
difference between the two states, ΔE = E2 - E1 = hν (the Bohr condition). However,
there are other “rules” governing such transitions, as we shall see.
To illustrate the general nature of the problem, think about an electron in a Bohr orbit of
the hydrogen atom and its interaction with electromagnetic radiation; e.g., a light wave.
We ask the following question: Given the fact that the electron is in a particular state at
time t = 0, what is the probability that it will be found in some other stat at a later time t?
To answer this question, we recall from the postulates of QM that if the functions ψ 0j are
eigenfunctions of the time-independent Schrödinger equation
Ĥ 0 ψ 0j
 E 0j ψ 0j
(1)
then the time-dependent wavefunction is of the form
ψ oj
 ψ 0j e iE jt / 
(2)
Each Ψ j0 is a solution of the time-dependent Schrödinger equation,
Ĥ 0 Ψ 0
 i
Ψ 0
t
(3)
Although each Ψ j0 represents a stationary state of the system, the most general solution
of (3) would be the superposition state
Ψ0



j0
a j Ψj0
(4)
Chem 342 - Quantum Mechanics of Light Absorption
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where for normalization it is necessary that

a *j a j
 1
(5)
j
Each product a *j a j is a measure of the probability that the system will be found in a
particular state j with energy Ej. Thus, although Ψ0 is a general function, we can
determine fro the coefficients in the expansion (4) the probability that we will find the
system in a particular state j. This is how we will solve this problem.
To be specific, we suppose at time t=0 that the system (that is, the electron in orbit
about the nucleus) is in a state labeled by the quantum numbers n1, ℓ1, m  1 which we
denote as 1. Schematically, this can be represented as
Thus, at time t=0,
Ψ0
 a 1 Ψ10 , a 1
 1.
(6)
all other terms in the expansion being zero. Next, we turn on the perturbation, which is
imagined to be a light wave oscillating at some particular frequency ν:
Ex
 E 0x cos [2π(z / λ  νt )]
(7)
We ask; what is the probability at some later time that the system will be found in some
other state? Clearly, this can be determined by finding the values of the coefficients in
the expansion
Ψ  a 1 Ψ10  a 2 Ψ20  a 3 Ψ30  
(8)
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Schematically, this can be represented as
Note that

a *j a j
 1.0.
j
To make life simple, we assume that the system has only two states, 1 and 2, and that a1 =
1.0 at time t=0. We further assume that the frequency of our light wave obeys the Bohr
frequency condition
ν 
E 2  E1  / h
(9)
Clearly, the wavefunction of the system
Ψ  a 1 Ψ10  a 2 Ψ20
(10)
must, at any time t, obey the time-dependent Schrödinger equation
Ψ
 2 2

Ĥ Ψ  
  V  Ψ  i
t
 2m

(11)
The potential energy function V now includes, in addition to the Coulomb potential
-Ze2/r, a perturbation
Ĥ    ex E 0x cos2π(z / λ  νt )
(12)
which represents the interaction of the electric field of the light wave with an electric
dipole oriented in x-direction. If we substitute Eqs. (10) and (12) into Eq. (11), and
simplify, we find
 ex E 0x cos 2π(z / λ  νt ) a 1 ( t ) Ψ10  a 2 ( t ) Ψ20 
da ( t )
da ( t ) 

 i  Ψ10 1  Ψ20 2 
dt
dt 

(13)
Chem 342 - Quantum Mechanics of Light Absorption
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where the time dependence of the coefficients a1 and a2 is explicitly shown. Taking
advantage of the orthogonality of the unperturbed functions Ψ j0 , we next multiply both
sides of Eq. (13) by the function Ψ2* and intgrate, obtaining as a result
iλ
da 2 (t )
dt

  e E 0x a 1 (t ) cos(2πνt )
Ψ2* x̂ Ψ1 dτ
(14)
The quantity da2(t)/dt represents the rate of transitions from the state 1 to the state 2.
Solving for a2(t) yields short times t (where a1(t) ~ 1, and Ψ2* , Ψ1  ψ*2 , ψ1 )
a 2 (t ) ~ 
eE 0x t
i

ψ*2 x̂ ψ1 dτ
(15)
Hence, the probability of finding the atom in the state 2 at time t is
a *2 a 2
~
e 2 (E 0x ) 2 t 2
2

2
ψ*2 x̂ ψ1 dτ .
(16)
This is called Fermi’s Golden Rule.
We may use this rule to derive the selection rules for any electric dipole transition. The
quantity

ψ *2 e x̂ ψ1 dτ  μ 12
(17)
is called the transition moment. Clearly, μ12 must be nonzero if the transition is allowed;
otherwise, we say that the transition is forbidden. The dipole moment operator for two
particles which have charges ±e is μ(r) = er, with components
μx
 μ (r ) sin θ cos φ
μy
 μ (r ) sin θ cos φ
μz
 μ (r ) cos φ
(18)
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Thus, to determine the selection rules for electronic transitions in the hydrogen atom, we
must evaluate
μ x 
μ y  
μ z 


0
sin θ 
R n μ R n ' ' r dr   Θ m sin θ  Θ ' ' sin θ dθ
cos θ   m
0
π
2
2π


0
cosψ 
Φ*m  sin ψ  Φ m ' dφ
 1 
(19)
These three factors give, respectively, the selection rules for n, ℓm and mℓ which are
Δn  anything ; Δ   1 ; Δm
Example…
 0,  1
(20)
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Example …shown here is the “term scheme” for atomic sodium, showing the allowed
transitions at not too high resolution. (From Born, Atomic Physics, Hafner, 1962).
A useful reference for this material is Atkins & De Paula, Chapter 12, pp. 358-359.