tuesday, january 29, 2008 nanostructures in biodiagnostics and gene therapy tuesday, january 29,...
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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 University
Regents Lecture: Aerosols in the Atmosphere: From the Ozone Hole to Climate Change Physical 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 Seminar 11 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
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)
In most situations the population in a state of energy Ei is given by a Boltzmann distribution.
Tki
begn ii
Expressed as a probability by dividing by N
a
kTi
i
akT
i i ii
Z
egp
ZegnN
i
i
Occupation in the ith level
i
j
k
Partition Function
Connection to Spectroscopy:
nAneZ
A
ApA
nAnA
n
kT
a
nnn n
nn
n ˆ1 /
This implies we’ve already calculated (or measured) all the energies.
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.
neneandenen
nAenZ
A
nHn
kTkTHkTkTH
n
kT
a
n
nn
n
//ˆ//ˆ
/ ˆ1
ˆ
As long as n is an eigenstate of H. Then,
n
a
kTH
nAZ
enA ˆ
/ˆ
and
ATrA
nAnA
Z
e
n
a
kTH
ˆˆ
ˆˆ
ˆ/ˆ
Define a new operator, the density operator
When we write Tr(A) (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.
For
translationrotationvibrationelectronic
Za = ZtranslationZrotationZvibrationZelectronic
Energies add wavefunctions multiplyEnergies add partition functions multiply
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 statesa,b,c represent distinct molecules
Na
N
i
kT
j
kT
i
kTensemble
Ze
eeZ
ai
bjai
,
,,
This assumes that molecules a,b,c are distinguishable as individuals.
However, when they are indistinguishable this results in overcounting.
Molecule a b cMicrostate A i j k
Microstate B j i k
The two states are identical. N! extra if count all of them.Thus
!N
ZZ
Na For indistinguishable particles
Interacting Particles
The above treatment is only valid for non-interacting (wavefunctions not overlapping) particles.
What happens when wavefunctions do overlap?
Total wavefunction, , for N particles is a function of all N coordinates
(1,2,3, . . ., N)
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.
| (1,2,3, . . ., N)|2 = | (2,1,3, . . ., N)|2
Thus
(1,2,3, . . ., N) = ±(1,2,3, . . ., N)
Fermions: -1•half-integral spins (electrons)• changes sign on particle exchange•Fermi-Dirac statistics•cannot share quantum states•Pauli-exclusion principle
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)
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
VNi
ii T
ZkTp
where
nE
,
2 ln
+
+
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.
BandE
Ê and B can be derived from the scalar potential and the vector potential Ã.
AB
t
AE
In free space with no charges .0The 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:
trkeAA
t
A
cA
cosˆ
1
0
2
2
22 – propagation vector (wavevector)
k = 2/c; =2 = unit vector
A0 = amplitude
k
e
We can get the magnetic field from
trkeAt
AE
sinˆ0
The amplitudes of the fields are
trkekAAB
sinˆ0
c
AB
AE
00
00
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 0E02 J/m3
This is the average energy density.
The intensity (irradiance) is the energy/time/area
I=uc
The Poynting vector S points along the direction of propagation with magnitude equal to the power/unit area.
2002
1
20
EceIS
BEcS
Polarization
Polarization is a key property because of a variety of conservation equations where the angular momentum of the photon participates.
cytiEcytkEE xz /cosˆ/cosˆ 00
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 45ºIf Ex = Ez and =90º right circularly polarized If Ex ≠ Ez and =90º elliptical
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:
tkrr
kqdtrE
cos4
sin,
0
20
Radiation from a charge distribution (molecule)
2
032
2420
32
sin
rc
qdI
Note: fourth power in quadratic in dipole moment (qd)no radiation parallel to the dipole
Effect of radiation on a charged particle
Lorenz Law
BEqF
v
A Hamiltonian that includes the effects of radiation as implied by the Lorenz law makes the substitution:
Aepp
Effect of radiation on a charged particle
AAm
e
m
iV
mH
Vim
Vpm
H
2AA
2
e
2ˆ
Ae2
1Ae
2
1ˆ
22
2
22
For weak fields, neglect terms quadratic in A and use:
AAA2
'ˆˆAe
2ˆ
02
2
HHm
iV
mH
Some experimental considerations:
Right and left circular polarizationLinear polarization can be resolved into rcp and lcp componentsMaterials that are birefringent e.g calcite, sapphire transmit act differently along their crystallographic axesQuarter and half-wave plates.