l19 - Quantum statistics Einstein and radiation Lasers...

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Unformatted text preview: Quantum statistics Einstein and radiation Lasers Lasers: NIF/LLNL The Fermi sea Quantum statistics: review • Fundamental equation: n ( E ) = g ( E ) f ( E ) dE . • Number n ( E ) of particles is density of available states g ( E ) times probability of occupying those states f ( E ) . • Maxwell-Boltzmann: non-interacting, non-integer occupancy of states. Ideal gas. f MB ( E ) = 1 exp [ E / k B T ] (1) • Bose-Einstein: integer occupancy of 0, 1, 2 . . . . Photons in a cavity, lasers. f BE ( E ) = 1 exp [ E / k B T ]- 1 (2) • Fermi-Dirac: integer occupancy of either 0 or 1. Electrons; Pauli exclusion principle. Fermi energy is E F = h 2 2 m e ( 3 N 8 π V ) 2 / 3 , where N / V is density of valence electrons. f FD ( E ) = 1 exp [( E- E f ) / k B T ] + 1 (3) Quantum statistics Einstein and radiation Lasers Lasers: NIF/LLNL The Fermi sea f MB and f BE Here’s a plot of f MB ( E ) = 1 exp [ E / k B T ] and f BE ( E ) = 1 exp [ E / k B T ]- 1 at room temperature for T = 300K. 0.00 0.0 0.5 1.0 1.5 Probability of occupying available states f 0.02 0.04 0.06 0.08 Energy (eV) For T =300K, or k B T =0.026 eV f BE : Bose-Einstein f M B : M a x w e ll-B o ltz m a n n Quantum statistics Einstein and radiation Lasers Lasers: NIF/LLNL The Fermi sea Einstein and radiation • This is done in Serway Sec. 12.7. Consider a two-level system, with energies E 2- E 1 = h ν , and populations N 1 and N 2 : N 1 ,E 1 N 2 ,E 2 h ν Spontaneous emission Absorption A 21 N 2 B 12 N 1 ρ ( ν ) • Spontaneous emission : the rate at which we lose electrons from state N 2 is proportional to the number of electrons in that state: parenleftbigg dN 2 dt parenrightbigg spont =- A 21 N 2 (4) • Absorption : the rate at which we pump electrons up to state N 2 is proportional to the number of electrons in state N 1 and the photon density ρ ( ν ) : parenleftbigg dN 1 dt parenrightbigg abs =- parenleftbigg dN 2 dt parenrightbigg abs =- B 12 N 1 ρ ( ν ) (5) Quantum statistics Einstein and radiation Lasers Lasers: NIF/LLNL The Fermi sea Einstein and radiation II • Einstein proposed a third process: N 1 ,E 1 N 2 ,E 2 h ν Spontaneous emission Stimulated emission Absorption A 21 N 2 B 12 N 1 ρ ( ν ) B 21 N 2 ρ ( ν ) • Stimulated emission : we can also drive transitions from state 2 to state 1 in proportion to the population of state 2 and the photon density ρ ( ν ) : parenleftbigg dN 2 dt parenrightbigg stim =- B 21 N 2 ρ ( ν...
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l19 - Quantum statistics Einstein and radiation Lasers...

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