Transcript lecture #4 ppt

Focus the Nation
Global Warming Solutions for
America
Jan 31, 2008
Location: The International
House, Berkeley
Morning Session
8:30 am – 9:00
Coffee/Tea
am
9:00 am – 9:15 Welcome - Nathan Brostrom, Vice Chancellor am
Administration, UC Berkeley
9:15 am –
10:15 am
The Future of the Planet - Climate Science for Citizen
Action
(Moderated by Daniel McGrath of the Berkeley Institute of
the Environment)
With Professors Inez Fung, Bill Nazaroff, Bill Collins,
John Chiang and Nathan Sayre
10:20 am 10:50 am
Keynote: Fran Pavley, Former California Assemblywoman
11:00 am –
12:00 pm
Climate Solutions – Institutional, Local, Individual
(Moderated by Prof. Dan Kammen, Energy and Resources
Group)
With panelists from Cal Climate Action Partnership, City of
Berkeley, Berkeley Energy and Resources Collaborative,
Chancellor’s Advisory Committee on Sustainability, the
Berkeley Institute of the Environment, and 1Sky
Session
12:30 pm –
12:45 pm
12:45 pm –
3:20 pm
Performance by the Golden Overtones
Solutions for America
(student led parallel breakout sessions on topic areas)
Organized and moderated by Students for a Greener
Berkeley
Policy Debates:
1) State, Regional, and Local Leadership on Sustainable
Transportation
2) Presidential Candidate Debate: Who is Prepared to
Make the US a Climate Action Leader?
Local Action:
1) Reducing Cal's Climate Impact
2) "Reaching Beyond the Choir": an innovative approach to
reducing our individual carbon footprint
3) The Green Initiative Fund – Speed Dating
3:20 pm - 3:30
pm
Remarks by Vice Provost Cathy Koshland, UC Berkeley
3:30 pm – 4:00 Concluding Keynote: Dr. Steven Chu, Nobel Laureate and
pm
Director of the Lawrence Berkeley National Laboratory
Quantization of the radiation field
Unification of QM energy levels
and the idea of absorption and
emission in discrete steps
For a single transverse wave (vector potential)

i k r t 
A(r, t )  A0eˆ e


For a cavity where many waves exist
simultaneously, we have a superposition of
waves of different k.

A(r , t ) 
1
 oV
qk (t )  qk e
uˆk (r )  eˆk e
it
 q (t )uˆ (r )
k
k ,eˆ
k
text includes both
polarizations in k as
do these notes after
this point
qk the vector amplitude
ik r
ek the spatial polarization
The total energy (classical) is obtained by averaging the energy density over the
cavity
H rad 
0
2

E c B
2
2
2


dr
Using the superposition formula and the relations
We derive:


A
E
t


B   A
H rad   q   q   p   q
1
2
k
2
k
2
2
k
1
2
k
2
k
2
2
k
Recall H.O. Hamiltonian

2
d
1 kx2 
H  T V   

2 dx 2
2
2
1
2
p
2
x
  2 x2
H rad   q   q   p   q
1
2
k
2
k
2
2
k
1
2
k
2
k
2
2
k

By analogy to H.O define raising and lowering operators (creation and annihilation)
1
k qk  ipk  raising
b 
2k

k
1
k qk  ipk  lowering
b 
2k

k
Slight rearrangement of the constants for consistency with other treatments, but
identical to a+ and a- as define for H.O.

 n  1 n  1
  n  n 1
b k nk 

b k nk
k
k
k
k
Here n is the occupation number
That is the number of photons of
wavevector k
By analogy to H.O define raising and lowering operators (creation and annihilation)
1
k qk  ipk  raising
b 
2k

k
1
k qk  ipk  lowering
b 
2k

k
Slight rearrangement of the constants for consistency with other treatments, but
identical to a+ and a- as define for H.O.

 n  1 n  1
  n  n 1
b k nk 

b k nk
k
k
k
k
|nk> =|0> ?
Using the commutation relation and the definitions of b+ and b-:
qk , pk   i k ,k '
The radiation Hamiltonian can be rewritten

H rad   p   q   k b b  b b
1
2
 
k k 
bb
k
2
k
2
1
2
2
k
 
k k
1 b b

H rad   k b b 
k
 
k k
1
2

k

 
k k
1
ˆ
  k nk  2
k
 
k k


 
k k
b b
 
k k
Is the occupation number operator. With integer eigenvalues
that specify the occupancy of wavevector and polarization k.

k
b b nk  b
nk nk  1  nk nk nk
n
4
En (h/2π)
The quantum theory of
radiation associates a QM H.O.
with each mode of the field.
4½
3
3½
2
2½
1
1½
0
½
n photons being excited in a
mode of the radiation field.
Now back to our cavity:
"Blackbody radiation" or "cavity radiation"
refers to an object or system which
absorbs all radiation incident upon it and
re-radiates energy which is characteristic
of this radiating system only, not
dependent upon the type of radiation
which is incident upon it. The radiated
energy can be considered to be
produced by standing wave or resonant
modes of the cavity which is radiating.
We expect that as T increases the body will glow more brightly
and the average emitted frequency will increase.
Modes in the cavity:.
Cube of size L. Allow k vectors with of L=nl in each of the 3
Cartesian axes (x,.y,z)
2
n 2
 kx 
lx
L
x2 for polarization
The mode density (number of modes/unit volume) is calculated as
The ratio of the “volume in k—space” to the “volume per mode”
k 3
k 3 L3 8 3 L3
Nk 
x 2( polarizati on) 

3
2
3
3c 3
2 / L 
4
3
d (Nk )
8 2
/ Volume  3
d
c
Mode density per unit frequency
and per unit volume
Solution: Assume energy levels of En=nh and that the
probability of occupancy follows the Boltzmann
distribution.
g ne
pn 
Z

n
kT

e

e
nh
kT

nh
kT
n
How many photons per mode?
n  n n pn  1
e
 / kT

1
Then the average energy for a blackbody
oscillator is
E
 nh e
n
e
n

nh

kT
nh
kT
h

e
nh
kT
1
h>>kT drives the ratio to zero
How does photon number fluctuate in this cavity?
Pn  n  / 1  n 
n
1 n
The probability of exactly measuring photons in
an optical radiation field depends on the
characteristics of the light source.
The radiation of a thermal light source shows a
Bose-Einstein count distribution.
The radiation of an ideal single-mode laser
shows a Poisson count distribution.
http://demonstrations.wolfram.com/PhotonNumberDistributions/
Next week: perturbation Theory. CH4