Statistical mechanics/Density Matrix

1. Tuesday, January 29, 2008Nanostructures in Biodiagnostics and Gene TherapyOrganic Chemistry 11 a.m.-12 p.m. |  Pitzer Auditorium, 120 Latimer Hall Speaker: Professor Chad Mirkin, Director of the Institute of Nanotechnology, George B. Rathmann Professor of Chemistry, Professor of Medicine, Professor of Material Sciences & Engineering, Northwestern UniversityRegents Lecture: Aerosols in the Atmosphere: From the Ozone Hole to Climate ChangePhysical Chemistry 4-5 p.m. |  Pitzer Auditorium, 120 Latimer Hall Dr. Doug Worsnop, Director, Center for Aerosol and Cloud Chemistry, Aerodyne Research, Inc. Wednesday, January 30, 2008Bayer Lecture in Biochemical Engineering: Engineering Challenges of Protein Formulations, 4-5 p.m. |  Pitzer Auditorium, 120 Latimer Hall Professor Theodore W. Randolph, Gillespie Professor of Bioengineering, University of Colorado at Boulder Thursday, January 31, 2008Graduate Research Seminar11 a.m.-12 p.m Pitzer Auditorium, 120 Latimer Hall Chemically Functionalized AFM Tips as a Tool for Studying Cell Biology: Sonny Hsiao, Ph.D. Student with Professor Matthew Francis' Research Group Investigating Lipid Membranes at the Liquid/Solid InterfaceProfessor Paul Cremer, Dept. of Chemistry, Texas A & M UniversitySurface Science & Catalysis, 1:30-2:30 p.m. | Lawrence Berkeley National Laboratory, Bldg. 66 Auditorium Graduate Research ConferenceGraduate Research Conference | 4-5 p.m. |  Pitzer Auditorium, 120 Latimer Hall Detection and Analysis of Polycyclic Aromatic Hydrocarbons (PAHs) with the Mars Organic Analyzer: Amanda Stockton, Ph.D. Student Observing the Weekend Effect on NOx from Space: Implications for Emissions and Chemistry: Ashley Russell, Ph.D. Student Friday, February 1, 2008Catalytic Activation of Carbon-Hydrogen Bonds using Ruthenium (II)Seminar: Inorganic Chemistry | February 1 | 4-5 p.m. | Pitzer Auditorium, 120 Latimer Hall Professor Brent Gunnoe, Dept. of Chemistry, North Carolina State University

2. Statistical mechanics/Density Matrix Statistical mechanics is the connection of properties of individual molecules with properties of an ensemble. Most experiments probe a volume containing large numbers of molecules—an ensemble. Spectroscopic measurement of the turnover rate of a single enzyme. Xie et al. (Harvard)

4. In most situations the population in a state of energy Ei is given by a Boltzmann distribution. k Occupation in the ith level Expressed as a probability by dividing by N j Partition Function i

5. Connection to Spectroscopy: The average value of A is given by the product of the expectation value of A and the probability of finding the molecule in a given state. Note value of A in state n, didn’t diagonalize in A but in H. Here each n unique, some n with identical energy. This implies we’ve already calculated (or measured) all the energies.

6. As long as n is an eigenstate of H. Then, and

7. Define a new operator, the density operator r When we write Tr(rA) (defined as adding up all of the diagonal elements of the operator acting on a basis set), there is no specific reference to a basis set. The trace is independent of representation (basis set). One can apply any unitary transformation you wish (change of basis set—maintaining complete and orthonormal set) and the trace stays the same.

8. Energies add wavefunctions multiply Energies add partition functions multiply For e = etranslation + erotation + evibration + eelectronic Za = ZtranslationZrotationZvibrationZelectronic

9. Bose-Einstein/Fermi-Dirac Statistics If the ensemble consists of multiple non-interacting molecules then the total energy is the sum of energies of the individual molecules. i,j,k represent states a,b,c represent distinct molecules This assumes that molecules a,b,c are distinguishable as individuals.

10. However, when they are indistinguishable this results in overcounting. The two states are identical. N! extra if count all of them. Thus For indistinguishable particles

11. Interacting Particles The above treatment is only valid for non-interacting (wavefunctions not overlapping) particles. What happens when wavefunctions do overlap? Total wavefunction, Y, for N particles is a function of all N coordinates Y(1,2,3, . . ., N)

12. Interacting Particles Since the probability distribution can’t depend on labels for indistinguishable particles, the exchange of any two labels must give the same result. | Y(1,2,3, . . ., N)|2 = | Y(2,1,3, . . ., N)|2 Thus Y(1,2,3, . . ., N) = ±Y(1,2,3, . . ., N)

13. Bosons: +1 • integral spin number • no sign change on particle exchange • Bose-Einstein Statistics • Bosons can share quantum states • Photons are Bosons: Spin=1 (lasers) • Fermions: -1 • half-integral spins (electrons) • Y changes sign on particle exchange • Fermi-Dirac statistics • cannot share quantum states • Pauli-exclusion principle

14. Z, the partition function. At low T, kT<< E1 Za =g0. All thermodynamic functions can be calculated once you know Z (CHEM 120B) e.g. Total Energy, E of a system

16. + + - - Classical electromagnetic radiation Maxwells equations describe the relations between an electric field and a magnetic field Transverse wave: E perpendicular to B and in phase. Both perpendicular to the direction of propagation.

17. Ê and B can be derived from the scalar potential  and the vector potential Ã. In free space with no charges . The vector potential A is of the most interest because it appears in the Hamiltonian for charged particles in a field. It obeys the wave equation: • – propagation vector (wavevector) • k = 2p/l = w/c; w=2pn • = unit vector • A0 = amplitude

18. We can get the magnetic field from The amplitudes of the fields are

19. Energy density is the square of the amplitude (B or E) and if we average this over a cycle: <E2>=1/2 E02 u=1/2 e0E02 J/m3 This is the average energy density. The intensity (irradiance) is the energy/time/area I=uc

20. The Poynting vector S points along the direction of propagation with magnitude equal to the power/unit area.

21. Polarization Polarization is a key property because of a variety of conservation equations where the angular momentum of the photon participates. Here we are adding two transverse waves moving in the +y direction where  is a phase shift of the x-component. If Ex = Ez and  =0º polarized at 45º If Ex = Ez and  =90º right circularly polarized If Ex≠ Ez and  =90º elliptical

22. Radiation from a charge distribution (molecule) At a large distance from a static charge distribution, it can be viewed as generating an electric field that can be described as a Taylor series: E= charge + dipole + quadrupole + … Often we can truncate after the first couple of termns. For oscillating charges (q) the same logic applies and the most important term is the dipole oscillation. If we assume harmonic oscillation, at distances (r) large compared to the charge separation (d), and at angle  between r and the dipole direction, the field is:

23. Radiation from a charge distribution (molecule) Note: fourth power in  quadratic in dipole moment (qd) no radiation parallel to the dipole

24. Effect of radiation on a charged particle Lorenz Law A Hamiltonian that includes the effects of radiation as implied by the Lorenz law makes the substitution:

25. Effect of radiation on a charged particle For weak fields, neglect terms quadratic in A and use:

26. Some experimental considerations: Right and left circular polarization Linear polarization can be resolved into rcp and lcp components Materials that are birefringent e.g calcite, sapphire transmit act differently along their crystallographic axes Quarter and half-wave plates.