Fermi’s Golden Rule
Time Dependent Perturbation Theory
Time dependent Schrödinger Equation
Take
time independent time dependent� will treat as time dependent perturbation
H0 time independent - solutions are
complete set of time independent orthonormal eigenfunctions of H0.
time dependent phase factors
Copyright – Michael D. Fayer, 2018
To solve
Expand
These terms equal. Unperturbed problem.�They cancel.
Substitute expansion
used derivative product rule
Canceling gives
Copyright – Michael D. Fayer, 2018
Have
Left multiply by
Therefore,
Eigenkets of time independent Hamiltonian
are orthonormal.
Exact to this point.�Set of coupled differential equations.�In Time Dependent Two State Problem (Chapter 8)
two coupled equations
Copyright – Michael D. Fayer, 2018
Approximations
Usually start in a particular state
Dealing with weak perturbation.�System is not greatly changed by perturbation.
Assume:
Time independent.
The probability of being in the initial state � never changes significantly.
The probability of being in any other state never gets much bigger than zero.
Copyright – Michael D. Fayer, 2018
With these assumptions:
No longer coupled equations
Excellent approximation in many common experiments.
UV/Vis spectrometer, FT-IR spectrometer, Fluorometer
Grazing Collision of an Ion and a Dipolar Molecule - Vibrational Excitation
M - molecule with� dipole moment
1. Molecule, M, weakly phys. absorbed on surface.� Not translating or rotating. (Example, CO on Cu surface.)
2. Dipole moment points out of wall.� Interaction with wall very weak; can be ignored.�3. When not interacting with ion – vibrations harmonic.�4. M has δ– side to right.
x
t
t
2
1
b
a
surface
+
+
I+
M
δ+
δ–
Copyright – Michael D. Fayer, 2018
Positively charged ion, I+, flies by M.�I+ starts infinitely far away at �Passes by M at t = 0.�I+ infinitely far away at
At any time, t,� I+ to M distance = a.
b = distance of closest approach (called impact parameter).�V = Ion velocity.
M
δ+
δ–
Copyright – Michael D. Fayer, 2018
Ion flies by molecule� Coulomb interaction perturbs vibrational states of M.
Model for Interaction
δ– end of M always closer to I+ than positive end of M.
Bond stretch energy lowered� δ– closer to I+.� δ+ further from I+.
Bond contracted opposite� Energy raised.
M
δ+
δ–
Copyright – Michael D. Fayer, 2018
Qualitatively Correct Model
Ion causes cubic perturbation of molecule.� Correct symmetry, odd.
Strength of Interaction� Inversely proportional to square of separation.� Coulomb Interaction of charged particle and dipole.
Neglects orientational factor - � but most effect when ion close - angle small.
Strength of interaction time dependent� because distance is time dependent.
M
δ+
δ–
Copyright – Michael D. Fayer, 2018
Time Dependent Perturbation
q = a constant (size of dipole, etc.)�x = position operator for H.O.
M starts in H.O. state . �Want probabilities of finding it in states after I+ flies by.� Time Dependent Perturbation Theory
M - I+ separation�squared. Ion - dipole�interaction.
M
δ+
δ–
Copyright – Michael D. Fayer, 2018
Take perturbation to be small.� Probability of system being in .
Probability of system being in .
Therefore,
Zeroth order time�dependent kets.
For this problem
ket�bra
Copyright – Michael D. Fayer, 2018
Substituting
Doesn't depend on H.O. coordinate,
take out of bracket.
Multiply through by dt and integrate.
Need to evaluate time independent and time dependent parts.
Copyright – Michael D. Fayer, 2018
Time independent part
Because x3 operates on � must have more raising operators than lowering. � Can't lower past �Can't have lowering operator on right.
Only terms in red survive.
Copyright – Michael D. Fayer, 2018
Then
Therefore,� m must be 1 or 3.�� Perturbation will only cause scattering to m = 1 and m = 3 states.� Only c3 and c1 are non-zero. (Higher odd powers of x � included in interaction – � population in higher odd� energy levels.)
Copyright – Michael D. Fayer, 2018
Time dependent part
Integral of odd function.
Substituting
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Putting the pieces together
Copyright – Michael D. Fayer, 2018
Probabilities
using
using
Probabilities are function of velocity.
P1 and P3 go to 0.�Must be maximum in between.
Copyright – Michael D. Fayer, 2018
Maximum value of as function of V.
Similarly
frequency of transition times�distance of closest approach
Copyright – Michael D. Fayer, 2018
frequency of transition times�distance of closest approach (impact parameter)
Note dependence on impact parameter.
Understanding maximum probabilities as function of V.
Calculate angular frequency when ion near molecule.
M
θ
b
a
d = Vt
I
+
When ion very close to point of closest approach
Copyright – Michael D. Fayer, 2018
For 0 → 1 transition, velocity for max. prob. is� V = ωb.
Then
Angular velocity – change in angle per unit time.
Copyright – Michael D. Fayer, 2018
Angular velocity
Looks like charged particle moving by M with angular velocity ω.
Produces E-field changing at� frequency ω.
On resonance efficiently induces transition.
For 0 → 3 transition, velocity for max. prob. is� V = 3ωb
Again on resonance.
Copyright – Michael D. Fayer, 2018
are eigenstates of the time independent Hamiltonian, .
These are eigenstates in the absence of coupling.
Fermi’s Golden Rule�an important result from time dependent perturbation theory
s0
s1
hν
Dense manifold of vibrational states of ground state – S0.
1012 to 1018 states/cm-1
Radiationless relaxation competes with fluorescence.
Many problems in which an initially prepared state is� coupled to a dense manifold of states.
First consider a pair of coupled state.
, where f = final and i = initial.
i
f
From time dependent perturbation theory
H′ is the time dependent piece of the Hamiltonian that couples the eigenstates.
Copyright – Michael D. Fayer, 2018
The time dependence of H′ is given by
The time dependent perturbation is 0 for t < 0, and a constant for t ≥ 0.
The probability of being in the final state is:
Copyright – Michael D. Fayer, 2018
The probability of finding the system in the final state
Using
Probability of being in the final state �as a function of t and ΔE.
Only good when small.
Copyright – Michael D. Fayer, 2018
Copyright – Michael D. Fayer, 2018
Probability of being in the final state F.
Must be small – time dependent perturbation theory
Probability must be small to use time dependent perturbation theory.
Take very short time limit.
Exact solution from time
dependent two state problem,
Chapter 8
Expand exact solution for short time.
Same at short time.
z = 20 cm-1
t = 50 fs
Square of zeroth order spherical Bessel function.
ΔE (cm-1)
This is a plot of the final probability at a single time
with z and t picked to keep final max probability low
as required to use time dependent perturbation theory.
Copyright – Michael D. Fayer, 2018
To this point, initial state coupled to a single final state.
To obtain result for initial state coupled to dense manifold of final states,
make following assumptions – very good for many physical situations.
2. The coupling bracket ��� can vary with f through the manifold of final states.� Take them all to be equal to z, which is now some average value� for the final states couplings to the initial state.
Probability of being in the manifold�of final states. Area under Bessel function.
Copyright – Michael D. Fayer, 2018
1. Most of the transition probability is close to ΔE = 0; ΔE2 in denominator. � Therefore, take the density of states, ρ, which is a function of energy, � to be constant with the value at ΔE = 0.
Let
Then
Using
Copyright – Michael D. Fayer, 2018
The transition probability per unit time is
Fermi’s Golden Rule
Rate constant for gain in probability in manifold of final states�equals rate constant for loss of probability from initial state.
Initial state can decay to zero without violating time dependent perturbation�theory approximation because so many states in final manifold that none�gain much probability. Need not consider coupling among states in manifold.
Copyright – Michael D. Fayer, 2018
Loss of probability from initial state per unit time a constant, k,�then
Probability of being in the initial state (population)�decays exponentially.
Radiative contribution, fluorescence, spontaneous emission,�kr (Chapter 12, Einstein A coefficient).
Fluorescence decay
τ = fluorescence lifetime
Nonradiative relaxation contribution to decay of �excited state population, knr, from Fermi’s Golden Rule.
s0
s1
hν
heat
Copyright – Michael D. Fayer, 2018