1. Transition Probabilities of Atoms and Molecules

2. Einstein’s analysis: Consider transitions between two molecular states with energies E 1 and E 2 (where E 1 < E 2 ). E ph is an energy of either emission or absorption. f is a frequency where E ph = hf = E 2 − E 1. If stimulated emission occurs: The number of molecules in the higher state (N 2 ) The energy density of the incoming radiation (u(f)) the rate at which stimulated transitions from E 2 to E 1 is B 21 N 2 u(f) (where B 21 is a proportional constant) The probability that a molecule at E 1 will absorb a photon is B 12 N 1 u(f) The rate of spontaneous emission will occur is AN 2 (where A is a constant) Spontaneous and stimulated emission

3. Once the system has reached equilibrium with the incoming radiation, the total number of downward and upward transitions must be equal. In the thermal equilibrium each of N i are proportional to their Boltzmann factor. In the classical time limit T → ∞. Then and u(f) becomes very large. The probability of stimulated emission is approximately equal to the probability of absorption. Stimulated Emission and Lasers

4. Solve for u(f), or, use Eq. (10.12), This closely resembles the Planck radiation law, but Planck law is expressed in terms of frequency. Eqs.(10.13) and (10.14) are required: The probability of spontaneous emission (A) is proportional to the probability of stimulated emission (B) in equilibrium. Stimulated Emission and Lasers

5. Laser: An acronym for “light amplification by the stimulated emission of radiation” Masers: Microwaves are used instead of visible light. The first working laser by Theodore H. Maiman in 1960 helium-neon laser

6. The body of the laser is a closed tube, filled with about a 9/1 ratio of helium and neon. Photons bouncing back and forth between two mirrors are used to stimulate the transitions in neon. Photons produced by stimulated emission will be coherent, and the photons that escape through the silvered mirror will be a coherent beam. How are atoms put into the excited state? We cannot rely on the photons in the tube; if we did: 1) Any photon produced by stimulated emission would have to be “used up” to excite another atom. 2) There may be nothing to prevent spontaneous emission from atoms in the excited state. The beam would not be coherent. Stimulated Emission and Lasers

7. Use a multilevel atomic system to see those problems. Three-level system 1) Atoms in the ground state are pumped to a higher state by some external energy. 2) The atom decays quickly to E 2. The transition from E 2 to E 1 is forbidden by a Δℓ = ±1 selection rule. E 2 is said to be metastable. 3) Population inversion: more atoms are in the metastable than in the ground state

8. Stimulated Emission and Lasers After an atom has been returned to the ground state from E 2, we want the external power supply to return it immediately to E 3, but it may take some time for this to happen. A photon with energy E 2 − E 1 can be absorbed. result would be a much weaker beam This is undesirable because the absorbed photon is unavailable for stimulating another transition.

9. Stimulated Emission and Lasers Four-level system 1) Atoms are pumped from the ground state to E 4. 2) They decay quickly to the metastable state E 3. 3) The stimulated emission takes atoms from E 3 to E 2. 4) The spontaneous transition from E 2 to E 1 is not forbidden, so E 2 will not exist long enough for a photon to be kicked from E 2 to E 3.  Lasing process can proceed efficiently.

10. Stimulated Emission and Lasers The red helium-neon laser uses transitions between energy levels in both helium and neon.

11. The magnetic dipole selection rules are, then: (1) No change in electronic configuration; (2) Parity is unchanged; (3) ∆J = 0, ±1; (4) ∆MJ = 0, ±1; (5) ∆J = 0 together with ∆MJ = 0 is not allowed; in particular, J = 0 ↔ 0 is not allowed; (6) ∆L = 0; (7) ∆S = 0. electric dipole selection rules for a single electron: (1) ∆L = ±1, ∆M = 0, ±1; (2) ∆S = 0, ∆MS = 0. electric dipole selection rules for many electron atoms are, then: (1) Only one electron changes its nl state; (2) Parity must change; (3) ∆J = 0, ±1; (4) ∆MJ = 0, ±1; (5) J = 0 ↔ 0 is not allowed; (6) ∆L = 0, ±1; (7) L = 0 ↔ 0 is not allowed; (8) ∆S = 0; where J ≡ L+S is the total orbital plus spin angular momentum Selection rules

20. Oxygen spectrum

21. Selection rules for vibrational versus rotational-vibrational Raman sp ectra Q-branch:Weak and for diatomic molecule not allowed Q-branch:allowed

22. Influence of nuclear spins on the rotational structure HFS is not treated here In thermal equilibrium a hydrogen molecule gas is a mixture of para to ortho in the ratio 1:3 The rotational spectrum can have no transitions with ΔJ= ±1and therefore no allowed transitions at all In contrast rotational Raman transitions with ΔJ= ±2 are allowed They belong alternatively to para and ortho states

23. Nuclear statistics Antisymmetric with exchange of the nuclei(nuclear spins) symmetric with exchange of the nuclei(nuclear spins) The odd rotational eigenfuctions with J=1,3,5…change their sign. Negative parity, antisymmetric The even rotational eigenfuctions with J=0,2,4…do not change their sign.Positive parity,symmetric

24. Figure 9-16 p333

25. Why does Bose-Einstein Condensation of Atoms Occur? Rb atomEric Cornell and Carl Wieman Na atomWolfgang Ketterle______ Consider boson and fermion wave functions of two identical particles labeled “1” and “2”. For now they can be either fermions or bosons: Nobel Price 2001 + symmetric =boson - antisymmetric=fermion Identicalprobability density the same = nonzero probability occupying the same state favors to be in the lower states for Bose-Einstein Conclusion :Solutions: Proof: