Transcript lecture #3 ppt

Tuesday, January 29, 2008
Nanostructures in Biodiagnostics and Gene Therapy
Organic 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, 2008
Bayer 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, 2008
Graduate 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 Interface
Professor Paul Cremer, Dept. of Chemistry, Texas A & M University
Surface Science & Catalysis, 1:30-2:30 p.m. | Lawrence Berkeley National Laboratory, Bldg. 66 Auditorium
Graduate Research Conference
Graduate 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, 2008
Catalytic 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.
k
Occupation in the ith level
ni  g i e
j
 i kbT
Expressed as a probability by dividing by N
N  i ni  i g i e

i

i
kT
 Za
i
g i e kT
pi 
Za
Partition Function
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.

A
nn
 n An
Here each n unique, some n
with identical energy.
A  n pn Ann
1
A
Za
e
n
 n / kT
n Aˆ n
This implies we’ve already calculated (or measured) all the energies.
 n n Hˆ n
1
A
Za

n
n e  n / kT Aˆ n
As long as n is an eigenstate of H. Then,
ne
 Hˆ / kT
 ne
 n / kT
and
and
A  n n
e
 Hˆ / kT
Za
Aˆ n
e
 Hˆ / kT
n  e  n / kT n
Define a new operator, the density operator 
ˆ 
e
 Hˆ / kT
Za
A  n n ˆAˆ n
 
A  Tr ˆAˆ
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.
Energies add wavefunctions multiply
Energies add partition functions multiply
For
  translation + rotation + vibration + electronic
Za = ZtranslationZrotationZvibrationZelectronic
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
Z ensemble  i e

 e
i

  i ,a kT
 i ,a kT N
e
  j ,b kT
j
 Z a 
N
This assumes that molecules a,b,c are distinguishable as individuals.
However, when they are indistinguishable this results in overcounting.
Molecule
a
b
c
Microstate A
i
j
j
i
k
k
Microstate B
The two states are identical. N! extra if count all of them.
Thus

Za 
Z
N
N!
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, Y, for N particles is a function of
all N coordinates
Y(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.
| Y(1,2,3, . . ., N)|2 = | Y(2,1,3, . . ., N)|2
Thus
Y(1,2,3, . . ., N) = ±Y(1,2,3, . . ., N)
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
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
E  n
where
  ln Z 
   pi i  kT 

 T  N ,V
i
2
Classical electromagnetic radiation

E

and B
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.
Ê and B can be derived from the scalar potential  and the vector potential Ã.


A
E   
t


B   A
In free space with no charges
  0 .
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:

 1  A
2
 A 2 2
c t
 

A  A0 eˆ cos k  r  t
2


k
– propagation vector (wavevector)
ê
= unit vector
k = 2p/l  /c;

A0 = amplitude
=2pn

 

A
E
 A0 eˆ sin k  r  t
t


We can get the magnetic field from
  

 


B    A   A0 k  eˆ sin k  r  t
The amplitudes of the fields are
E0  A0
 A0
B0 
c

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.



2
S   0c E  B
S  I  ce0 E
1
2
2
0
Polarization
Polarization is a key property because of a variety of
conservation equations where the angular momentum
of the photon participates.

E  Ez0kˆ cost  y / c + Ex0iˆ cost  y / c +  
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
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:
qd 0 k sin 
E r , t  
coskr  t 
4p0 r
2
Radiation from a charge distribution (molecule)

qd 0   4 sin 2 
I
2
32p c  0 r
2 3
2
Note: fourth power in 
quadratic in dipole moment (qd)
no radiation parallel to the dipole
Effect of radiation on a charged particle
Lorenz Law


  
F  q E + v B

A Hamiltonian that includes the effects of radiation as implied by the
Lorenz law makes the substitution:



p  p  eA
Effect of radiation on a charged particle




 2
 2
1 
1
ˆ
H
p  eA + V 
 i  eA + V
2m
2m


2
2 
 


i

e
e
2
Hˆ 
 +V +
A + A +
AA
2m
2m
2m
For weak fields, neglect terms quadratic in A and use:
 



  AY + A  Y   2 AY 
 



i

e
2
Hˆ 
 +V +
A  Hˆ 0 + Hˆ '
2m
m
2
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.