Embed Size (px)
Citation preview
Focus the NationGlobal Warming Solutions for
America
Jan 31, 2008
Location: The International House, Berkeley
Morning Session
8:30 am – 9:00 am
Coffee/Tea
9:00 am – 9:15 am
Welcome - Nathan Brostrom, Vice Chancellor - 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
Afternoon Session
12:30 pm – 12:45 pm
Performance by the Golden Overtones
12:45 pm – 3:20 pm
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 Transportation2) Presidential Candidate Debate: Who is Prepared to Make the US a Climate Action Leader?
Local Action: 1) Reducing Cal's Climate Impact2) "Reaching Beyond the Choir": an innovative approach to reducing our individual carbon footprint3) 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 pm
Concluding Keynote: Dr. Steven Chu, Nobel Laureate and 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
trkieeAtrA ˆ),( 0
rikkk
tikk
ekkk
o
eeru
eqtq
rutqV
trA
ˆ)(ˆ
)(
)(ˆ)(1
),(ˆ,
For a single transverse wave (vector potential)
For a cavity where many waves exist simultaneously, we have a superposition of waves of different k.
qk the vector amplitude
ek the spatial polarization
text includes both polarizations in k as do these notes after this point
The total energy (classical) is obtained by averaging the energy density over the cavity
Using the superposition formula and the relationsWe derive: AB
t
AE
rdBcEH rad
222
20
2222
12222
1k
kkk
kkrad qpqqH
Recall H.O. Hamiltonian
2222
122
22
21
2 xpkxdx
dVTH x
2222
12222
1k
kkk
kkrad qpqqH
By analogy to H.O define raising and lowering operators (creation and annihilation)
lowering2
1
raising2
1
kkk
k
k
kkk
k
k
ipqb
ipqb
Slight rearrangement of the constants for consistency with other treatments, but identical to a+ and a- as define for H.O.
1
11
kkk
kkk
nnnb
nnnb
k
k Here n is the occupation numberThat is the number of photons of wavevector k
By analogy to H.O define raising and lowering operators (creation and annihilation)
lowering2
1
raising2
1
kkk
k
k
kkk
k
k
ipqb
ipqb
Slight rearrangement of the constants for consistency with other treatments, but identical to a+ and a- as define for H.O.
1
11
kkk
kkk
nnnb
nnnb
k
k |nk> =|0> ?
',, kkkk ipq
Using the commutation relation and the definitions of b+ and b-:
The radiation Hamiltonian can be rewritten
2
1
2
1
2
12222
1
ˆ
1
kk
kkkk
krad
kkkk
kkkkk
kkk
krad
nbbH
bbbb
bbbbqpH
Is the occupation number operator. With integer eigenvalues that specify the occupancy of wavevector and polarization k.
kk bb
kkkkkkkkk nnnnnbnbb 1
n
4
3
2
1
0
En (h/2π)
4½
3½
2½
1½
½
The quantum theory of radiation associates a QM H.O. with each mode of the field.
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=nin each of the 3 Cartesian axes (x,.y,z)
The mode density (number of modes/unit volume) is calculated as
The ratio of the “volume in k—space” to the “volume per mode”
x2 for polarization
L
nkx
x
22
3
33
2
33
3
33
4
3
8
3)(2
/2 c
LLkonpolarizatix
L
kNk
3
28/)(
cVolume
d
Nd k
Mode density per unit frequencyand per unit volume
Solution: Assume energy levels of En=nh and that the probability of occupancy follows the Boltzmann distribution.
n
kT
nh
kT
nh
kTn
n
e
e
Z
egp
n
How many photons per mode?
11/
kTnn epnn
1
kT
nh
n
kT
nh
kT
nh
n
e
h
e
enhE
Then the average energy for a blackbody oscillator is
h>>kT drives the ratio to zero
How does photon number fluctuate in this cavity?
nnn nnP 11/
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
