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

Perturbation Theory: Energies and Intensities

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

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.

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.

S1

S1 vibrational level Lifetime (ns) Toluene 85.7 ± 1.1 00 Toluene 85.7 ± 1.1 Toluene–Ne 77.2 ± 2.6 Toluene–Ar 42.5 ± 2.0 371 Toluene 70.7 ± 1.9 Toluene–Ne 74.7 ± 4.2 Toluene–Ar 82.6 ± 2.2 131/241251 Toluene 69.2 ± 3.7 Toluene–Ne contaminated Toluene–Ar 82.1 ± 0.8

Butyl-nitrite

Weak 300-400nm Strong ~220 nm

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

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

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.

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|

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

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

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

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

H’(t)=E-M field

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

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

Next time: Fermi’s Golden Rule