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Lecture Topics Oct. 28 What we just finished
Wavefunctions of multiple particle system, identical particles,
Differences between classical particles, bosons, fermions
2 identical particles in a box for classical, bosons, fermions
Document 10A_identical_particles
Today: Statistical Physics (Chap. 10)
Microstates & macrostates & probabilities
Probabilities for classical, fermion, boson
Density of states
Document 10B_statphys
Next lecture (Chap. 10)
Maxwell-Boltzmann distribution (energy, velocity). Applications
Document 10C_MB_statistics PHY3101: Chapter 10<br>
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HITT Points If you have 0 HITT points it most likely means that you didn’t register your remote. PHY3101: Chapter 10 Period 2
1 remote: 841646
2 remote: 846729
3 remote: 835761
4 remote: 836227
5 remote: 852175
2 Particles in Potential Box: Dramatic Difference! PHY3101: Chapter 10 D S A (n1,n2) = (1,2)<br>
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Microstates and Macrostates Microstate example: gas system described per molecule
6N microstates (3 position, 3 velocity)
Macrostate example: gas system described by overall properties
Gas total energy, pressure, temperature, entropy PHY3101: Chapter 10<br>
Consider Energy Levels (Distinguishable Particles) PHY3101: Chapter 10 Q = 6 units of energy shared among N = 5 particles 10 macrostates: A – J What is the distribution of individual energies for the 5 particles?
Next slide<br>
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Energy Distribution (Counting from Diagram) PHY3101: Chapter 10 210 microstates × 5 particles = 1050 individual energies
E = 0: 420 = 4×5 + 3×20 + 3×20 + 2×30 + 2×60 + 3×10 + 2×10 + 1×20 + 1×30
E = 1: 280 = 1×20 + 2×30 + 1×60 + 3×20 + 2×30 + 4×5
E = 2: 175 = 1×20 + 1×60 + 3×10 + 2×30 + 1×5
E = 3: 100 = 1×60 + 2×10 + 1×20
E = 4: 50 = 1×20 + 1×30
E = 5: 20 = + 1×20
E = 6: 5 = 1×5
1050 total Statistical physics axiom: Each microstate is equally likely (Next slide)<br>
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Particle Energies: Boltzmann Energy Distribution PHY3101: Chapter 10 Approximately exponential Boltzmann energy distribution.
But ~exactly exponential when N ~ 1023, E huge<br>
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Indistinguishable Particle Energy Distributions PHY3101: Chapter 10 Major differences in counting indistinguishable particle states:
Each macrostate has 1 microstate (they are indistinguishable)
Spin ½ fermions can have only 2 particles / energy level, so states with >2 particles are not possible (3 of 10 states allowed)<br>
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Energy Distribution for Bosons/Fermions PHY3101: Chapter 10 Bosons Fermions Easy to calculate from figure assuming 1 microstate/macrostate<br>
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Distribution Functions Summary Distribution function is probability of a single particle having energy E at a given temperature T PHY3101: Chapter 10 μ = “chemical potential” (somewhat T dependent)
For photons, μ = 0. Basis of Planck radiation formula
Next slide: Plots of distributions for a given T<br>
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PHY3101: Chapter 10<br>
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Density of States For a fixed set of N particles and total energy E, we determined the particle energy distribution (MB, BE, FD)
But we still don’t know how many energy states are between energy E and E + dE.
We define the density of states g(E)
g(E)dE = number density of particles between E and E + dE
We work this out now PHY3101: Chapter 10<br>
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Density of States (2) We imagine a very large cube containing the physical system
For particles to be contained in the box, the wavefunction defining the available states is
(These integers are enormous for macroscopic systems) PHY3101: Chapter 10<br>
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Density of States (3) Let’s define a spherical region defined by PHY3101: Chapter 10 Define g(n) dn = # states/volume # spin states V # states in dn shell But Can now calculate g(E)<br>
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Density of States (4) PHY3101: Chapter 10 # states/volume<br>
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Particle Number Density Total number density between E and E + dE
where f(E) = # particles/energy state
We determined f(E) for 3 cases: distinguishable, bosons, fermions
Example: Energy expectation value
Examples next time PHY3101: Chapter 10 fMB, fBE, fFD Total # density<br>
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PHY3101: Chapter 10 g(E) Electron f(E) Electron Product<br>
