(i) Given 3 distinguishable particles each of which can be...

(i) Given 3 distinguishable particles each of which can be in any of 4 states with (different) energies $E_{1}, E_{2}, E_{3}$, and $E_{4}$, (a) what is the total number of ways of distributing the particles amongst the states?, (b) how many states of the system have total energy $E_{1}+E_{2}+E_{4}$ ? (ii) Repeat (i) for 3 electrons instead of distinguishable particles.

Key Concepts

Fermi-Dirac Statistics

Fermi-Dirac statistics apply to indistinguishable particles with half-integer spin, such as electrons. These statistics are characterized by the rule that no two fermions can occupy the same quantum state simultaneously, which is a direct consequence of the Pauli exclusion principle.

Microstates and Macrostate Energy

A microstate is a specific configuration of particles among various energy levels. When every arrangement or distribution corresponding to a fixed total energy (a macrostate) is considered, the counting of microstates determines the statistical properties and probability of that macrostate.

Pauli Exclusion Principle

This principle states that in a system of fermions, such as electrons, each quantum state can be occupied by at most one particle. It plays a critical role in determining the allowed configurations of particle distributions in quantum systems, particularly affecting how microstates are counted for systems of identical particles.

Distinguishability

This concept refers to whether the particles in a system can be told apart on the basis of their labels or identities. In problems involving distinguishable particles, each particle is considered unique, which affects the way microstates are counted because swapping two particles leads to a new configuration.

Combinatorial Analysis

Combinatorial techniques are used to count the number of ways particles can be distributed among available states. This analysis often involves basic counting principles such as the multiplication rule and permutations, particularly when each particle has multiple states available.

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