p. 43
MIT Department of Chemistry
5.74, Spring 2004: Introductory Quantum Mechanics II
Instructor: Prof. Andrei Tokmakoff
FERMI’S GOLDEN RULE
We have calculated the probability of observing the system in a state k after applying a
. Often we are interested in transition probability not to an individual eigenstate,
perturbation to
but a distribution of eigenstates. Often the set of eigenstates form a continuum of accepting states,
for instance, vibrational relaxation or ionization.
Transfer to a set of continuum (or bath) states forms the basis for a describing irreversible
relaxation. Qualitatively, you expect deterministic, oscillatory feedback between discrete quantum
states. However, the amplitude of one discrete state coupled to a continuum will decay due to
destructive interferences between the oscillating frequencies for each member of the continuum.
So we are interested in calculating transition probability to a distribution of final states: Pk .
Pk = bk
2
Probability of observing amplitude in discrete eigenstate of H0
Density of states—units in  1 E  , describes distribution of final
k
states—all eigenstates of H0
ρ(Ek ):
If we start in a state
, the total transition probability is a sum of probabilities Pk = ∑ Pk . We are
k
just interested in the rate of leaving
and occupying any state k . or for a continuous
distribution:
k
Ek − El
P k = ∫ dEk ρ (E k ) Pk
l
For a constant perturbation:
P k = ∫ dE k ρ ( E k ) 4 Vk
2
sin 2 ( ( E k − E ) t / 2
Ek − E
2
)
p. 44
Let’s make two assumptions:
1)
ρ ( Ek )
ρ(E k ) doesn’t vary much with frequency.
There are many final states: continuous.
Also, t is relatively long.
2) 2π / t
Pk
Ek − El
The matrix element Vk is invariant across the final states. These assumptions allow those variables to be factored out of integral
Pk = ρ ( E k ) Vk
2
∫
+∞
−∞
dE k 4
sin 2 ( E k − E ) t / 2
( Ek − E )
2
We have chosen the limits −∞ → + ∞ since ρ(Ek ) is broad relative to Pk . Using
sin 2 a∆
∫−∞ d∆ ∆ 2 = aπ , with a = t /
+∞
Pk =
2π
ρ ( E k ) Vk
2
t
The total transition probability is linearly proportional to time. Often, for instance in relaxation
processes, we will be concerned with the transition rate, w k :
∂P k
∂t
2π
ρ ( E k ) Vk
wk =
wk =
2
This is Fermi’s Golden Rule. Note rates independent of time. It is very common for chemical rate
processes: matrix element squared + D.O.S.
Remember that Pk is centered sharply at Ek = E . So we may write ρ(Ek = E
commonly in terms of δ (Ek − E ) :
) or more
p. 45
wk =
wk =
2π
ρ ( Ek = E
2π
Vk
2
) Vk
2
δ ( Ek − E
)
w k = ∫ dE k ρ ( E k ) w k
Range of validity:
Vk << ω k
For discrete states we saw:
Also Pk never varies much from initial values.
Pk = w k ( t − t 0 )
t <<
1
wk
However, transition probability must be sharp compared to ρ(Ek ).
ρ ( Ek )
2π / t
Pk
Ek − El
t >> / ∆E k
∆E >> w k
∆ωk >> w k
∆E k = ∆ωk
p. 46
Golden Rule Rate for Harmonic Perturbations
If we want to calculate the transition rate to a continuum of final states induced by harmonic
perturbation, we follow derivation of F.G.R. from before.
Only absorption: Pk =
Vk
2
4
2
sin 2 ( ωk − ω) t 2 
2
( ωk − ω) / 2
P k = ∫ dE k ρ ( E k ) Pk
= ∫ dωk ρ ( ωk ) Pk
=
wk =
=
π
Vk
2 2
2
ρ ( ωk = ω + ω)( t )
∂P k
π
= 2
Vk
∂t
2
π
Vk
2 2
2
2
ρ ( ωk = ω + ω)
ρ ( ωk = ω)
or more commonly:
wk =
π
Vk
2 2
2
δ ( ωk − ω)
If we include both the absorption and stimulated emission and neglect interferences (i.e., long
times):
wk =
=
π
Vk
2 2
π
Vk
2
2
2
δ
 ( ωk − ω) + δ ( ωk + ω) 
δ ( E k − E − ω) + δ ( E k − E + ω) 


abs: E k = E + ω
S.E. E k = E − ω
p. 47
Let’s look at this expression for the two sets of states {i } and
STIMULATED EMISSION
j
w ij =
i
j
2
w ij
w ji
=
π
Vij ρ ( E i )
2 2
ABSORPTION
i
since Vij = V ji
{ j } where ε j > ε i :
w ji =
2
π
Vji ρ ( E j )
2
2
2
ρ ( Ei )
ρ ( E j
)
detailed balance
The ratio of the rates upward and downward is given by the ratio of the density of states of the
photons in the electric field. (More commonly this is seen written for matter in the form that
relates the rates for discrete states to the thermal occupation of those states:
w ij / w ji = exp ( −β ωij ) .