PolarizationCircularly Polarized LightGeneral Aspects of SpectroscopyElectromagnetic RadiationLight is crossed electric and magnetic field that are oscillating in time.Changing electric field creates a magnetic field and a changing magnetic field creates an electric field, etc…BEt  2Ec Bt Light is self-propagating electromagnetic field.Electric field:  0E E sin kz t    Magnetic field:  0H H sin kz t    yxzk – wavevectork 2  – wavenumber1n =l – frequency2w= pn – phase (describes origin in a coordinate frame)The spatial and temporal oscillations are related by kc (relationship is always true for light)PolarizationPolarization gives the direction of the electric field.Circularly Polarized LightIn the most general formulation, the electric field rotates about the propagation axis.EView as photoncomes towardsyou.ELeft-circularly polarized Right-circularly polarized  0ˆ ˆE E sin kz t i i j     0ˆ ˆE E sin kz t i i j   1Photon has one unit of angular momentum.Lh- two polarizations  two values for ML: ML = 1- note ML = 0 is missing (no longitudinal field)Plane Polarized LightPlane polarized light is a linear combination of circularly polarized photons.Consider “animation” belowE-E+EyE-E+EyE-E+E = 0yE-E+EyE-E+EyBecause it is often simplest to consider molecules in a Cartesian coordinate system plane-polarized light is often very useful to probe the property of matter.Time-dependent Schrödinger EquationSo far in our study of quantum chemistry, we have been studying Hamiltonians and solutions to the Schrödinger equation that have been time-independent (that is, operators and wavefunctions do not change with time.The interaction of light with matter is time-dependent process, so let us briefly consider the time-dependent Schrödinger equation.H it  hThus, two operators exist for the energy.ˆE H, ˆE ithThe time-dependent wavefunctions are related to the time-independent wavefunctions via thefollowing solution.   niE tn nr, t r e h2Time-dependent Perturbation TheoryBackgroundIn searching for solutions to the time-dependent Schrödinger equation, the Hamiltonian is split into a time-independent portion and time-dependent portion. 0 1H(t) H H t Perturbation theory is now applied where the time-dependent portion of the Hamiltonian is considered the perturbation. The total wavefunction is constructed from a linear combinationof zeroth-order wavefunctions, just as in time-independent perturbation theory; however, the coefficients are allowed to be time dependent.           1 1 2 2 3 3a t t a t t a t t       LTwo-state wavefunctionGenerally, a time-dependent interaction involves only two states so that the total wavefunction is a linear combination of a lower energy state and a higher energy state.       1 1 2 2a t t a t t    The remainder of the perturbation analysis is similar to the analysis for first-order time-independent perturbation theory and it yields expressions for the time-dependent coefficients.     011 2i t12da t a tH t edt i h     012 1i t21da t a tH t edt ihwhere        1 112 1 2H t r H t r d   and2 10E E hAside: ( )112H is called a matrix element. The matrix element is analogous to an expectation value. A matrix element describes the probability of a system changing from state 1 to state 2 under the influence of the perturbation, ( )( )1H t. (Perturbation does not have to depend upon time.)The fact that the coefficients depend on each other reflects the idea that the while the perturbation is applied, the system oscillates between the two states.3Solution for wavefunction coefficientsTo find the a2 coefficient, solve the above differential equation and allow that the interaction is very small so that a1 remains approximately constant at a value of 1.If a2 is initially zero (i.e., a2(0) = 0), then    0t1i t2 1201a t H t e dtihLet       1 1 1i t i tH t 2H cos t H e e     Note the difference between 0 (a difference in energy levels of the quantum state) and  (the frequency of the perturbation (light))The probability that the system is in state 2 is the square of the a2 coefficient.     1 12212 21 02 22204H H sin 1 2 tP a       hFermi’s Golden RuleFermi’s Golden Rule relates how quickly a transition occurs between a lower energy level (1) and a higher energy level (2). The transition rate, 2 1W, is the time derivative of the probability of the state being in the higher energy level.22 1dPWdtStarting with the probability found in the previous section and after some sophisticated calculus and a few reasonable physical assumptions, Fermi’s Golden Rule can be stated as  2122 1 12 N 2dP 2W H Edt  hwhere N 2E is density of energy level near energy level 2.Key points of Fermi’s Golden Rule1. Transition rate depends on the square of the transition matrix element.- If spectroscopic transition, matrix element is transition dipole matrix element.2. Transition rate is independent of time.3. The higher the density of energy levels (more levels within a smaller energy range), the higher the transition rate.4Stimulated and Spontaneous Interactions of Dipole RadiationAbsorptionFor an interaction between light and matter to occur (absorption), the oscillating dipole electric field of the incident light must be able to induce a dipole within the matter.Absorption is always stimulated.Absorption is proportional to the power of incident light.f inumber of moleculesdNWabsorbed per unit timedt W NB N – number of moleculesB – Einstein absorption coefficient - energy density48 kT EmissionLight is emitted only when dipole in matter oscillates.Such an oscillation can be spontaneous or it can be stimulated with a photon.Stimulated emission is proportional to the power of the incident light.f inumber of moleculesdNWemitting per unit timedt f iW NB  B′ – Einstein