Consider a system of two atoms, each having ony 3 quantum...

Consider a system of two atoms, each having ony 3 quantum states of energies $0, \varepsilon$ and $2 \varepsilon$. The system is in contact with a heat reservoir at temperature $T$. Write down the partition function $Z$ for the system if the particles obey (a) Classical statistics and are distinguishable. (b) Classical statistics and are indistinguishable. (c) Fermi-Dirac statistics. (d) Bose-Einstein statistics.

Consider a system of two atoms, each having ony 3 quantum states of energies $0, \varepsilon$ and $2 \varepsilon$. The system is in contact with a heat reservoir at temperature $T$. Write down the partition function $Z$ for the system if the particles obey
(a) Classical statistics and are distinguishable.
(b) Classical statistics and are indistinguishable.
(c) Fermi-Dirac statistics.
(d) Bose-Einstein statistics.

Key Concepts

Fermi-Dirac Statistics

Fermi-Dirac statistics applies to fermions, which are particles that abide by the Pauli exclusion principle stating that no two fermions can occupy the same quantum state simultaneously. In building the partition function for fermions, it is essential to consider that each quantum state can be either unoccupied or occupied by a single fermion, which significantly alters the counting of microstates compared to classical or bosonic systems.

Bose-Einstein Statistics

Bose-Einstein statistics describes bosons, particles that do not obey the Pauli exclusion principle and can therefore occupy the same quantum state in unlimited numbers. When constructing the partition function for a bosonic system, the possibility of multiple occupancy of the same state is included, leading to a different statistical treatment than that used for fermions or distinguishable particles.

Distinguishability

Distinguishability is a key concept that affects the formulation of the partition function. If particles are distinguishable, each permutation is considered a separate microstate. In contrast, when particles are indistinguishable, permutations of particles among the same energy levels are not counted as different, and appropriate combinatorial corrections must be made in the partition function.

Microstates and Energy Levels

In statistical problems, a key step is the identification and enumeration of all possible microstates along with their energy levels. Each microstate corresponds to a specific arrangement of particles among the available energy levels, and the probability of each microstate is given by its Boltzmann factor. This concept is crucial for writing down a partition function that correctly accounts for the contributions from each configuration.

Partition Function

The partition function is a central concept in statistical mechanics that encapsulates all the statistical properties of a system in thermal equilibrium. It is defined as the sum over all possible microstates of the system of the Boltzmann factor, exp(??E), where ? is the inverse temperature and E is the energy associated with each microstate. This function is used to calculate thermodynamic quantities like energy, entropy, and free energy.

Classical Statistics

Classical statistics, often represented by Maxwell-Boltzmann statistics, assumes that particles are distinguishable and that the probability of a state is determined solely by its energy through the Boltzmann factor. When particles are regarded as distinguishable, each specific arrangement (permutation) is considered unique. However, when the particles are indistinguishable, a correction factor, such as dividing by N! (where N is the number of particles), is applied to avoid overcounting identical configurations.

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