Show all 26 classical distributions

How many ways can you distribute 9 units of energy among 6 identical, indistinguishable fermions?

Fermi-Dirac statistics differ dramatically from the classical Maxwell-Boltzmann statistics in that fermions must obey the Pauli exclusion principle. Considering the particles in this example to be electrons, a maximum of two particles can occupy each spatial state since there are two spin states each. Whereas there were 26 possible configurations for distinguishable particles, these are reduced to the 5 states which have no more than two particles in each state.

Evaluating the average occupancy of each energy state is much simpler than in the Maxwell-Boltzmann example since each macrostate has a weight of 1. The average occupancy is just the sum of the numbers of particles in a given energy state over all the 5 distributions divided by 5.

Evaluate the average for each energy state.Compare with Maxwell-Boltzmann example.
Index

Reference
Blatt
Ch. 11
 
HyperPhysics***** Quantum Physics R Nave
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The average distribution of 9 units of energy among 6 identical particles

EAverage
number
Maxwell-
Boltzmann
Average
number
Bose-
Einstein
Average
number
Fermi-
Dirac
0
2.143
2.269
1.8
1
1.484
1.538
1.6
2
0.989
0.885
1.2
3
0.629
0.538
0.8
4
0.378
0.269
0.4
5
0.210
0.192
0.2
6
0.105
0.115
0
7
0.045
0.077
0
8
0.015
0.038
0
9
0.003
0.038
0

For fermions, there are only 5 possible distributions of 9 units of energy among 6 particles compared to 26 possible distributions for classical particles. To get a distribution function of the number of particles as a function of energy, the average population of each energy state must be taken. The average for each of the 9 states is shown above compared to the results obtained by Maxwell-Boltzmann statistics and Bose-Einstein statistics .

Low energy states are less probable with Fermi-Dirac statistics than with the Maxwell-Boltzmann statistics while mid-range energies are more probable. While that difference is not dramatic in this example for a small number of particles, it becomes very dramatic with large numbers and low temperatures. At absolute zero all of the possible energy states up to a level called the Fermi energy are occupied, and all the levels above the Fermi energy are vacant.

Index

Reference
Blatt
Ch. 11
 
HyperPhysics***** Quantum Physics R Nave
Go Back