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Quantum Statistics:Applications

Quantum Statistics:Applications. Determine P(E) = D(E) x n(E) probability(E) = density of states x prob. per state electron in Hydrogen atom. What is the relative probaility to be in the n=1 vs n=2 level? D=2 for n=1 D=8 for n=2

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Quantum Statistics:Applications

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  1. Quantum Statistics:Applications • Determine P(E) = D(E) x n(E) probability(E) = density of states x prob. per state • electron in Hydrogen atom. What is the relative probaility to be in the n=1 vs n=2 level? • D=2 for n=1 D=8 for n=2 • as density of electrons is low can use Boltzman: • can determine relative probability • If want the ratio of number in 2S+2P to 1S to be .1 you need T = 32,000 degrees. (measuring the relative intensity of absorption lines in a star’s atmosphere or a interstellar gas cloud gives T) P460 - Quan. Stats. II

  2. 1D Harmonic Oscillator • Equally spaced energy levels. Number of states at each is 2s+1. Assume s=0 and so 1 state/energy level • Density of states • N = total number of “objects” (particles) gives normalization factor for n(E) • note dependence on N and T P460 - Quan. Stats. II

  3. 1D H.O. : BE and FD • Do same for Bose-Einstein and Fermi-Dirac • “normalization” varies with T. Fermi-Dirac easier to generalize • T=0 all lower states fill up to Fermi Energy • In materials, EF tends to vary slowly with energy (see BFD for terms). Determining at T=0 often “easy” and is often used. Always where n(E)=1/2 P460 - Quan. Stats. II

  4. Density of States “Gases” • # of available states (“nodes”) for any wavelength • wavelength --> momentum --> energy • “standing wave” counting often holds:often called “gas” but can be solid/liquid. Solve Scrd. Eq. In 1D • go to 3D. ni>0 and look at 1/8 of sphere 0 L P460 - Quan. Stats. II

  5. Density of States II • The degenracy is usually 2s+1 where s=spin. But photons have only 2 polarization states (as m=0) • convert to momentum • convert to energy depends on kinematics relativistic • non-realtivistic P460 - Quan. Stats. II

  6. Plank Blackbody Radiation • Photon gas - spin 1 Bosons - dervied from just stat. Mech. (and not for a particular case) by S.N. Bose in 1924 • Probability(E)=no. photons(E) = P(E) = D(E)*n(E) • density of state = D(E) = # quantum states per energy interval = • n(E) = probability per quantum state. Normalization: number of photons isn’t fixed and so a single higher E can convert to many lower E • energy per volume per energy interval = P460 - Quan. Stats. II

  7. Phonon Gas and Heat Capacity • Heat capacity of a solid depends on vibrational modes of atoms • electron’s energy levels forced high by Pauli Ex. And so do not contribute • most naturally explained using phonons - spin 1 psuedoparticles - correspond to each vibrational node - velocity depends on material • acoustical wave <---> EM wave phonon <---> photon • almost identical statistical treatment as photons in Plank distribution. Use Bose statistics • done in E&R Sect 11-5, determines P460 - Quan. Stats. II

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