Thermodynamic And Statistical Physics

The total energy of the system is given by . Since there are three distinguishable particles, and the first spin has , the probability of having state is calculated based on the number of favorable configurations for the spins.

Step 1: The probability depends on the number of available states and the magnetic moment associated with the spin. The system has three possible configurations, so the probability for each state is .

Step 2: The magnetic moment in the +z direction for the first spin is , resulting in the given configuration.
Thus, the correct answer is (A).

For a system of Fermions (particles obeying the Pauli exclusion principle, such as electrons), the chemical potential plays a crucial role in determining the energy distribution of the particles. The density of states increases with energy, and as the temperature increases, the particles are more likely to occupy higher energy states. At absolute zero, the chemical potential is the energy at which the Fermi-Dirac distribution function equals 1/2. As the temperature rises, more particles occupy higher energy states, which causes the chemical potential to decrease with increasing temperature. Therefore, the correct answer is (A).

We are given a system of two atoms in three quantum states with energies 0, , and . 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 is simply the sum over the Boltzmann factors of the states. The three states have energies , , and , so the partition function is: 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: 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: 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: This matches with Option (C) for BE: Z2. Thus, the correct match is:
- CD: Z3
- CI: Z4
- FD: Z1
- BE: Z2
Final Answer: (C)

The Helmholtz free energy is given by the formula: where is the partition function, is the temperature, and is the Boltzmann constant. For a quantum harmonic oscillator, the partition function is given by: where is the energy of the nth state, and . Substituting the energy expression into the partition function: This sum is a geometric series, and can be summed as: Now, the Helmholtz free energy is: This simplifies to: Thus, the correct expression for the Helmholtz free energy is given by option (B).