Focus the Nation Global Warming Solutions for America Jan 31, 2008 Location: The International House, BerkeleyThe 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 AssemblywomanFran Pavley 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 breakout sessions 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 pm Concluding Keynote: Dr. Steven Chu, Nobel Laureate and Director of the Lawrence Berkeley National LaboratoryDr. Steven Chu

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) For a cavity where many waves exist simultaneously, we have a superposition of waves of different k. q k the vector amplitude e k 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 relations We derive:

Recall H.O. Hamiltonian

By analogy to H.O define raising and lowering operators (creation and annihilation) Slight rearrangement of the constants for consistency with other treatments, but identical to a + and a - as define for H.O. 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) Slight rearrangement of the constants for consistency with other treatments, but identical to a + and a - as define for H.O. |n k > =|0> ?

Using the commutation relation and the definitions of b+ and b - : The radiation Hamiltonian can be rewritten

Is the occupation number operator. With integer eigenvalues that specify the occupancy of wavevector and polarization k.

n43210n43210 E n (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=n  in 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

Mode density per unit frequency and per unit volume

Solution: Assume energy levels of E n =nh and that the probability of occupancy follows the Boltzmann distribution. How many photons per mode?

Then the average energy for a blackbody oscillator is

h >>kT drives the ratio to zero

How does photon number fluctuate in this cavity?

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.

Next week: perturbation Theory. CH4