Lecture
Second-Order Transitions
Lecture Prepared by: Zhitao Kang
Reading:
Brennan Chapter 4.5
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Fermi’s golden rule for first-order transitions
Recall the transition rate of first-order transition is expressed by Fermi’s golden rule:
2
2π
(there is a band of energy for the final states);
W=
k V s ρ f (E s )
h
2
2π
Or: W =
k V s δ (Ek − Es ) (single state transition; energy conserved)
h
Where s is the initial state, k is the final state, V is the time-dependent perturbation,
ρ f (Es ) describes the final density-of-states.
Therefore, transition rate W depends on two terms:
kV s
2
: the coupling between the initial and the final states;
ρ f (Es ) : final density-of-states.
Forbidden transitions: If matrix elements of k V s vanishes, the transition between the
initial and the final states cannot occur.
Examples:
1. Some transitions between excited atomic states;
2. Optical transitions in certain semiconductors.
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Second-order transitions
¾Forbidden transitions cannot occur to first order, but they may occur to second
and higher order through some intermediate path.
¾For example, a transition can occur first from initial state s to an intermediate
state m followed by decay to the final state k.
¾The net result of a second-order transition is the same as in a direct transition
but the path is different.
s
k
Initial state
s
Intermediate state
m
Final state
First-order transition
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X
k
Second-order transition
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Different formalism pictures in Quantum Mechanics
¾ To understand how second or higher order transitions occur and to derive the
transition rate, a different but equivalent formalism can be used, which is called the
interaction picture.
¾ In the interaction picture both the state vectors and the operators exhibit time
dependence.
¾ The interaction picture is intermediate between the Schroedinger and the Heisenberg
pictures.
Operator
Schroedinger Equation:
State vector
HΨ = EΨ
Time independent
Time dependent
Schroedinger Picture
Heisenberg Picture
Interaction Picture
State vectors
Time dependent
Time independent
Time dependent
Operators
Time independent
Time dependent
Time dependent
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Equivalent Schroedinger equation in the interaction picture
How to write the Schroedinger equation in the interaction picture?
In Schroedinger picture, in the presence of some perturbation V the Schroedinger
equation can be written as:
(H 0 + V )Ψ (t ) = ih dΨ (t )
dt
Define a state vector Ψ′(t) that is the time-evolved form of Ψ(t) as:
Ψ′(t ) ≡ e
iH 0t
h
Ψ (t )
Take the differentiation of the above definition, we can construct the equation of the
motion for Ψ′(t):
iH 0t
dΨ′ iH 0 iHh0t
dΨ (t )
e Ψ (t ) + e h
=
dt
h
dt
Both sides of the above equation are multiplied by iħ,
iH 0t
iH 0t
dΨ ′
dΨ (t )
= − H 0e h Ψ (t ) + e h ih
ih
dt
dt
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Equivalent Schroedinger equation in the interaction picture
iH 0 t
iH 0 t
dΨ ′
dΨ (t )
ih
= − H 0 e h Ψ (t ) + e h ih
dt
dt
Insert the expression of Schroedinger equation into the second term at the ride side of the
above equation,
iH 0 t
iH 0 t
iH 0 t
dΨ ′
ih
= − H 0 e h Ψ (t ) + e h (H 0 + V )Ψ (t ) = e h V Ψ (t )
dt
Cont’d:
And from the definition: Ψ (t ) ≡ e
Therefore
− iH 0t
h
Ψ′(t )
iH 0t
− iH 0t
dΨ ′
= e h Ve h Ψ′(t )
ih
dt
iH 0t
h
− iH 0t
h
Define V′ as: V ′ ≡ e Ve
Then we got the equivalent Schroedinger equation in the interaction picture:
dΨ ′(t )
= V ′ Ψ ′(t )
dt
The operator V′ can be thought of as an effective Hamiltonian, which is time dependent.
Time dependence of operator V′ is determined by H0. Time dependence of state vector ψ is
determined by the perturbation.
ih
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Time-evolution operators in Schroedinger and interaction pictures
In the Schroedinger picture a linear operator T(t,t0) is defined as a timetranslation or time-evolution operator, which is used to derive the translation
rate for first-order transitions.
T(t,t0) translates the system in time from t = t0 to t = t, which is quantitatively
defined as:
Schroedinger picture Ψ (t ) = T (t , t0 )Ψ (t0 )
In the interaction picture, a time-evolution operator U(t,t0) is similarly defined,
which can be used for the derivation of the translation rate for second and
higher order transitions.
Interaction picture
Ψ′(t ) = U (t , t0 )Ψ′(t0 )
If U(t,t0) can be solved, then the transition probability and rate between the
initial state Ψ(t0) and the final state can be calculated.
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Time-evolution operators in Schroedinger and interaction pictures
What is the relationship between T(t,t0) and U(t,t0)?
Ψ′(t ) = U (t , t0 )Ψ′(t0 )
As defined previously, in interaction picture:
Insert the definition of Ψ′(t):
Times e
− iH 0t
h
Ψ ′(t ) ≡ e
e
iH 0t
h
iH 0t
h
Ψ (t ) = U (t , t0 )e
iH 0t0
h
Ψ (t0 )
at both the left and right sides:
Ψ (t ) = e
From the definition of T(t,t0):
Then we got:
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Ψ (t )
− iH 0t
h
U (t , t0 )e
iH 0t0
h
Ψ (t0 )
Ψ (t ) = T (t , t0 )Ψ (t0 )
U (t , t0 ) = e
iH 0t
h
T (t , t0 )e
}
− iH 0t0
h
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Solution of time-evolution operator U(t,t0) in interaction picture
To solve U(t,t0) in the interaction picture, first let we see how T(t,t0) is solved in the
schroedinger picture.
The time-translation operator T(t,t0) can be determined with the help from the timedependent Schroedinger equation, which is:
ih
dΨ (t ) − iH (t )
=
dt
Ψ (t )
h
dΨ (t )
= H (t )Ψ (t )
dt
Integrating with respect to time, we can write the above equation as:
dΨ (t ) − i
H dt
=
∫
∫
(
)
t
h
Ψ
Ψ (t 0 )
t0
Ψ (t )
t
Integrating and substituting in the bounds on the left-hand side yields, if H is time
independent,
 Ψ (t ) 
(t − t0 )
ln 
=
−
iH

h
 Ψ (t0 ) 
Which reduces to:
Ψ (t ) = Ψ (t0 )e
− iH (t −t 0 )
h
Definition of T(t,t0): Ψ (t ) = T (t , t0 )Ψ (t0 )
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}
T (t , t0 ) = e
− iH (t −t 0 )
h
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Solution of time-evolution operator U(t,t0) in interaction picture
Cont’d:
T (t , t0 ) = e
− iH (t −t 0 )
h
For small time increments (t-t0) ~εoperator T(t,t0) can be expressed by a Taylor series
expansion of the exponent as:
iHε
T (t + ε , t ) = 1 −
h
Because T(t,t0) is a linear operator:
T (t + ε , t0 ) = T (t + ε , t )T (t , t0 )
The derivative T(t,t0) of can be evaluated as:
dT (t , t0 )
T (t + ε , t0 ) − T (t , t0 )
= lim
ε →0
dt
ε
[T (t + ε , t ) − 1]T (t , t0 )
= lim
ε →0
=−
Then we get:
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ih
i
HT (t , t0 )
h
ε
dT (t , t0 )
= HT (t , t0 )
dt
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Solution of time-evolution operator U(t,t0) in interaction picture
From the similarity of the Schroedinger picture and interaction picture, we can get the
expression for U(t,t0) based on the result of T(t,t0) obtained previously:
Schroedinger picture
Interaction picture
V ′ Ψ ′(t ) = ih
Definition of
T(t,t0) and U(t,t0):
dΨ (t )
HΨ (t ) = ih
dt
Ψ (t ) = T (t , t0 )Ψ (t0 )
Ψ′(t ) = U (t , t0 )Ψ′(t0 )
T(t,t0) and U(t,t0):
ih
Schroedinger equation:
dT (t , t0 )
= HT (t , t0 )
dt
ih
dΨ ′(t )
dt
dU(t, t0 )
= V ′U (t, t0 )
dt
t
Integrating the expression of U(t,t0) U (t , t ) = U (t , t ) − i V ′(t ′)U (t ′, t )dt ′
0
0 0
0
h t0
with respect to time yields:
∫
t
U(t0,t0) is an identity operator:
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i
U (t , t0 ) = 1 − ∫ V ′(t ′)U (t ′, t0 )dt ′
h t0
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Solution of time-evolution operator U(t,t0) in interaction picture
Cont’d:
t
i
U (t , t0 ) = 1 − ∫ V ′(t ′)U (t ′, t0 )dt ′
h t0
This integral equation can be approximately solved as following using an iterative
procedure if the perturbation is small.
First the value of U(t′,t0) is approximated by U = 1. Substituting it into the above integral
equation we get the zeroth-order value for U(t,t0) :
t
i
U (t , t0 ) = 1 − ∫ V ′(t ′)dt ′
h t0
Substituting this zeroth-order U(t,t0) into the integral equation again we get the first-order
value of U(t,t0)
2 t
t
t
t
t′


i
i
i
i


U (t , t0 ) = 1 − ∫ V ′(t ′)1 − ∫ V ′(t ′)dt ′ dt ′ = 1 − ∫ V ′(t ′)dt ′ +  −  ∫ V ′(t ′)dt ′∫ V ′(t ′′)dt ′′
 ht

h t0
h t0
 h  t0
t0
0


By repeating this procedure we get higher orders of solution until the level of accuracy is
obtained. The iterative solution can be written as:
t
2 t
t′
i
 i
U (t , t0 ) = 1 − ∫ V ′(t ′)dt ′ +  −  ∫ V ′(t ′)dt ′∫ V ′(t ′′)dt ′′ + L
h t0
 h  t0
t0
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Probability of transition between the initial and final states
The probability amplitude of the electron will be found in a final state k at a later time t is
given by the kth coefficient Ck, which is determined from the overlap of the state Ψk on the
state Ψ′(t) as:
Ck = ∫ Ψk*Ψ′(t )d 3r
Ψ′(t) can be written as U(t,t0)Ψs , assuming the system is in state s initially. Therefore Ck
can be expressed in Dirac notation as:
C k = k U (t , t 0 ) s
Substituting the expansion for U(t,t0) into the above expression yields:
i
Ck = −
h
t
∫
t0
 i
k V ′ s dt′ +  − 
 h
first-order transition
probability amplitude
Insert
V′ ≡ e
iH 0t
h
Ve
− iH 0t
h
 i
C k (t ) =  − 
 h
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2 t
t′
t0
t0
∫ d t ′∫
k V ′(t ′ )V ′(t ′′ ) s d t ′′ + L
second-order transition
probability amplitude
and rewrite the second term as:
t′
∫ d t ′∫ d t ′′∑
t0
2 t
t0
ke
iH 0 t
h
Ve
− iH 0 t
h
m
me
iH 0 t
h
Ve
− iH 0 t
h
s
m
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Probability and rate of second-order transition
2 t
t′
iH 0t
iH 0t
− iH 0t
− iH 0t
 i
Ck (t ) =  −  ∫ dt ′∫ dt ′′∑ k e h Ve h m m e h Ve h s
 h  t0 t0
m
Using the property of exponential operators (Ψl(0) is an arbitrary definite eigenstate):
Cont’d
e
iH 0t
h
Ψl(0 ) = e
iEl( 0 )t
h
Ψl(0 )
The second-order Ck can be rewritten as:
 i
Ck (t ) =  − 
 h
2 t
t′
∫ dt′∫ dt′′∑
t0
t0
k V m mV s e
i ( Ek − Em ) t ′ i ( Em − E s ) t ′′
η (t ′+ t ′′ )
h
h
e
e
m
where eη (t ′+t ′′ ) is added to model a slow turn on of the potential. The fact η approaches 0
in the limit.
If we take t0 to approach - ∞, let η approach 0 in the limit, and perform the integrations
over t′ and t′′, square, and take the time derivative, we can write the second-order
transition rate as:
Ws→k
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2π
=
h
∑
m
k V m mV s
Es − Em
2
δ (Ek − Es )
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Probability and rate of second-order transition
In summary, the transition rate for second-order transition is derived by the following path:
T(t,t0)
Ws→k
U(t,t0)
2π
=
h
∑
m
Ck(t)
k V m mV s
Es − Em
Ws→k
2
δ (Ek − Es )
Several issues on second-order transition:
1. The transition probability is proportional to the product of two separate transition (s→m
and m→k) probabilities or matrix elements. Therefore the probability is far less than that
for a first order transition since it involves the product of two probabilities.
2. To determine the overall transition rate we must sum over the complete set of all
intermediate states m before squaring.
3. The intermediate states need not be real. The system can make a transition to an
intermediate virtual state and then to a final real state.
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Example of second-order transition
Second-order transition
E
First-order transition
E
phonon
s→k
photon
Optical recombination for
direct-gap semiconductors
k
photon
m→k
s→m
k
Optical recombination for
indirect-gap semiconductors
¾For indirect semiconductors such as Si and Ge, radiative transitions cannot occur to first
order since the matrix element of k V s vanishes because the momentum is not conserved.
¾The recombination can occur to second order through phonon-assisted process but the overall
probability is low. This is why Si and Ge exhibit weak luminescence efficiency compared to
direct-gap semiconductors.
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