Consider the distribution of a total of six energy units to a collection of five

Question

Consider the distribution of a total of six energy units to a collection of five identical Fermions (indistinguishable particles with half- integer spins). (a) How many macrostates are there? (b) Find the microstates, hence the multiplicity for each macrostate. (c) What is the average number of particles with a particular energy E? (d) What is the probability to measure any particular value of energy of a particle? (e) Plot the probability p(E) as a function of energy and try to find the expression for the Fermi-Dirac probability distribution.

Explanation / Answer

Let five identical fermions are A , A ,A , A ,A

Total energy =6E

The energy levels are 0E, 1E, 2E, 3E, 4E, 5E

Now as per Pauli's exclusion principle , no two fermions can occupy same energy state.

The distibution of fermions among these energy states are as follows

So from the above table we can observe, that if we place 1 particle in one energy state, then only 4 particles we can distibute over the energy levels for which the total enegy of the sysytem becomes six energy units.

So it is impossible to distribute 5 fermions over the energy level whose total energy is six units.

So no. of macrostate is zero.

0E 1E 2E 3E 4E 5E Total A A A A 6E

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