1. Tuesday, February 5, 2008
11 a.m.-12 p.m.
Rohm and Haas Lecture: Novel Materials for Drug Delivery and Tissue Engineering
Speaker: Professor Robert Langer, Massachusetts Institute of Technology
4-5 p.m
Self-Assembly of Nanostructured Materials
Speaker: Professor Bartosz Gryzbowski, Dept. of Chemical & Biological Engineering, Northwestern University

2. Perturbation Theory: Energies and Intensities

3. Approximate solutions
Variational Principle
 normalized; H exact
A matrix solution defined as a linear combination of the set of functions .
Energies, E0<E1<E2<…
Coefficients, c, describe the linear combination that minimizes the energies

4. Approximate solutions
2. Perturbation Theory
Assume we know the solutions to H0 as a set of E and 
This is a good approach if the series expansions above are converging.

5. The first order perturbation is a solved using the zero order eigenfunctions.
Note the denominator! Energies that are near affect each other most strongly.
Degenerate levels require some additional care in solutions.

8. S1

9. S1 vibrational level Lifetime (ns) Toluene 85.7 ± 1.1 00
Toluene  ± 1.1
Toluene–Ne  ± 2.6
Toluene–Ar  ± 2.0
371
Toluene  ± 1.9
Toluene–Ne  ± 4.2
Toluene–Ar  ± 2.2
131/241251
Toluene  ± 3.7
Toluene–Ne contaminated
Toluene–Ar  ± 0.8

14. Butyl-nitrite

15. Weak nm
Strong
~220 nm

18. Transition Strengths: Time dependent perturbation theory
Derive the rate of a driven transition (absorption or emission given) by Fermi Golden Rule as dependent on the transition dipole moment.
Time-dependent Schrodinger equation
S refers to Schrodinger representation; wavefunctions are time dependent
Where the wavefunction at t=0 is an eigenfunction of H. The time-dependent phase factor only affects the solution of time-dependent problems. For an operator, A, the expectation value is:
Operator evolves in time, instead of wavefunction

19. Which properties change with time and which are constant?
Solve using chain rule and use definition of commutator to simplify

20. At t=0
When As is not a function of time, the time dependence of the expectation value of A is given by the commutator of A and H.
e.g. position (x) and momentum (p) are time dependent, E is not.

21. First order solution: for H’(t) where H’(t) small compared to H0
With eigenvalues and eigenvectors of Ho denoted |n>
Substituting into the Hamiltonian and projecting onto the state <m|

22. where
Not useful in this form since every Cn depends on all others.

23. Assume at t=0,
in the state i and the state we follow is f.

24. Perturbation turned on at t=0
Integral in time determines the transition rate from i to f
if the exponential and Vfi(t) have a similar period of oscillation then the integral terms add
if they have different frequencies then they will cancel and the derivative will be zero (Pf(t)=0)
Time P Pi Pf 1

26. H’(t)=E-M field
Expand the wavelength exponential as

27. H’(t)=E-M field

28. H’(t)=E-M field
Ef<Ei Emission
Ef>Ei Absorption

29. H’(t)=E-M field D  = - fi Electric dipole transitions
The second term in the expansion k·r gives magentic dipole and electric quadrupole transitions. These terms are of order 10,000 times smaller.
D  = - fi

31. Next time: Fermi’s Golden Rule