1 / 7

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

Download Presentation

Quantum Statistics:Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related