- Fermions are particles that are identical and indistinguishable.
- Fermions include particles such as electrons, positrons, protons, neutrons, etc. They all have half-integer spin.
- Fermions obey the Pauli exclusion principle, i.e. each quantum state can only accept one particle.
- Therefore, for fermions Nj cannot be larger than gj.
- FD statistic is useful in characterizing free electrons in semi-conductors and metals.
For FD statistics, the quantum states of each energy level can be classified into two groups: occupied Nj and unoccupied (gj-Nj), similar to head and tail situation (Note, quantum states are distinguishable!) - For FD statistics, the quantum states of each energy level can be classified into two groups: occupied Nj and unoccupied (gj-Nj), similar to head and tail situation (Note, quantum states are distinguishable!)
- The thermodynamic probability for the jth energy level is calculated as
-
- where gj is N in the coin-tossing experiments.
- The total thermodynamic probability is
W and ln(W) have a monotonic relationship, the configuration which gives the maximum W value also generates the largest ln(W) value. - W and ln(W) have a monotonic relationship, the configuration which gives the maximum W value also generates the largest ln(W) value.
- The Stirling approximation can thus be employed to find maximum W
- There are two constrains
- Using the Lagrange multiplier
-
13.5 Bose-Einstein distribution - Bosons have zero-spin (spin factor is 1).
- Bosons are indistinguishable particles.
- Each quantum state can hold any number of bosons.
- The thermodynamic probability for level j is
- The thermodynamic probability of the system is
Finding the distribution function 13.6 Diluted gas and Maxwell-Boltzman distribution - Dilute: the occupation number Nj is significantly smaller than the available quantum states, gj >> Nj.
- The above condition is valid for real gases except at very low temperature.
- As a result, there is very unlikely that more than one particle occupies a quantum state. Therefore, the FD and BE statistics should merge there.
The above two slides show that FD and BE merged. - The above two slides show that FD and BE merged.
- The above “classic limit” is called Maxwell-Boltzman distribution.
- Notice the difference
- They difference is a constant. Because the distribution is established through differentiation, the distribution is not affected by such a constant.
Summary - Boltzman statistics:
- Fermi-Dirac statistics:
- Bose-Einstein statistics:
- Problem 13-4: Show that for a system of N particles obeying Maxwell-Boltzmann statistics, the occupation number for the jth energy level is given by
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