Question

A system of two atoms can be in three quantum states having energies 0, $\epsilon$ and $2\epsilon$. The system is in equilibrium at temperature \( T = (k_B\beta)^{-1} \). Match the following Statistics with the Partition function.

• A. CD: Z1, CI: Z2, FD: Z3, BE: Z4
• B. CD: Z2, CI: Z3, FD: Z4, BE: Z1
• C. CD: Z3, CI: Z4, FD: Z1, BE: Z2
• D. CD: Z4, CI: Z1, FD: Z2, BE: Z3

Hint:

For each statistical distribution, the partition function depends on whether particles are distinguishable or indistinguishable, and whether they obey Fermi-Dirac or Bose-Einstein statistics.

Answer 1

The Correct Option is C

We are given a system of two atoms in three quantum states with energies 0, \( \epsilon \), and \( 2\epsilon \). We need to match the partition function for different statistical distributions: Classical Distinguishable particles (CD), Classical Indistinguishable particles (CI), Fermi-Dirac (FD), and Bose-Einstein (BE). Step 1: Partition Function for CD (Classical Distinguishable Particles)
For classical distinguishable particles, the partition function \( Z_1 \) is simply the sum over the Boltzmann factors of the states. The three states have energies \( 0 \), \( \epsilon \), and \( 2\epsilon \), so the partition function is: \[ Z_1 = e^{-\beta 0} + e^{-\beta \epsilon} + e^{-\beta 2\epsilon} = 1 + e^{-\beta \epsilon} + e^{-2\beta \epsilon}. \] This corresponds to Option (C) for CD: Z3. Step 2: Partition Function for CI (Classical Indistinguishable Particles)
For indistinguishable particles, the partition function includes a factor of 1 for the ground state and two terms for the excited states. The correct partition function is: \[ Z_2 = 1 + e^{-\beta \epsilon} + 2e^{-2\beta \epsilon} + e^{-3\beta \epsilon} + e^{-4\beta \epsilon}. \] This matches with Option (C) for CI: Z4. Step 3: Partition Function for FD (Fermi-Dirac)
For Fermi-Dirac statistics, we must account for the Pauli exclusion principle. The partition function is given by: \[ Z_3 = 1 + 2e^{-\beta \epsilon} + 3e^{-2\beta \epsilon} + 2e^{-3\beta \epsilon} + e^{-4\beta \epsilon}. \] This matches with Option (C) for FD: Z1. Step 4: Partition Function for BE (Bose-Einstein)
For Bose-Einstein statistics, the partition function includes fractional terms as multiple particles can occupy the same state. The partition function is: \[ Z_4 = \frac{1}{2} e^{-\beta \epsilon} + \frac{3}{2} e^{-2\beta \epsilon} + e^{-3\beta \epsilon} + \frac{1}{2} e^{-4\beta \epsilon}. \] This matches with Option (C) for BE: Z2. Thus, the correct match is:
- CD: Z3
- CI: Z4
- FD: Z1
- BE: Z2
Final Answer: (C)