Question

Quantum statistics changes into classical statistics if: (Symbols have their usual meaning)

• A. \(\frac{g_i}{n_i} = 1\)
• B. \(\frac{g_i}{n_i} \gg 1\)
• C. \(\frac{g_i}{n_i} \ll 1\)
• D. \(\frac{g_i}{n_i} = 0\)

Hint:

The classical limit is the "non-crowded" limit. If there are many more available quantum "seats" (\(g_i\)) than there are particles (\(n_i\)) to sit in them, the particles are unlikely to interact in a way that requires quantum rules, so classical statistics work fine.

Answer 1

The Correct Option is B

Step 1: Understand the terms. - \(n_i\) is the number of particles in energy level \(i\). - \(g_i\) is the degeneracy of energy level \(i\), which is the number of available states at that energy level. - The ratio \(\frac{n_i}{g_i}\) is the "occupation index," representing the average number of particles per available quantum state.
Step 2: Characterize quantum vs. classical statistics. - **Quantum Statistics (Fermi-Dirac and Bose-Einstein):** These are necessary when the occupation index is not small. The distinguishability of particles (or lack thereof) and limits on occupation (Pauli exclusion principle for fermions) are critical. This corresponds to a high density of particles in the available quantum states. - **Classical Statistics (Maxwell-Boltzmann):** This is the limit where particles are treated as distinguishable and there are no restrictions on how many can occupy a state. This limit is valid when the occupation index is very small, meaning the particles are sparsely distributed among a large number of available states.
Step 3: Formulate the condition for the classical limit. The classical limit is reached when the occupation index is much less than 1: \[ \frac{n_i}{g_i} \ll 1 \] This is equivalent to saying that the number of available states \(g_i\) is much larger than the number of particles \(n_i\) that need to occupy them. \[ g_i \gg n_i \implies \frac{g_i}{n_i} \gg 1 \] This condition corresponds to a low-density, high-temperature gas where quantum effects are negligible.