An Interlude — Question 2

Question 2

Quantum statistics changes the number of two-particle occupations

Two identical noninteracting particles can occupy three distinct one-particle states of equal energy. Each state includes every internal quantum number. Compare spinless bosons with fermions whose spin state is fixed.

1. Count the allowed two-particle occupation patterns for each statistics.

2. For three particles in the same three states, repeat the count and explain the difference.

Question 2 - Solution

Quantum statistics changes the number of two-particle occupations

(a) Method and calculation

$$\boxed{\Omega_B(2,3)=\binom{3+2-1}{2}=\binom42=6}.$$$$\boxed{\Omega_F(2,3)=\binom32=3}\quad(n_i=0\text{ or }1).$$

(b) Interpretation and further calculation

$$\boxed{\Omega_B(3,3)=\binom53=10,\quad\Omega_F(3,3)=\binom33=1}.$$$$\sum_i n_i=N,\quad\boxed{n_i\ge0\text{ for bosons};\ n_i\le1\text{ for identical fermions}}.$$

Physical check

The count concerns indistinguishable occupation patterns, not labeled-particle arrangements. Degenerate energy does not make distinct quantum states identical.