The basic postulates of quantum statistics, specifically for quantum mechanics applied to identical particles, are:
Symmetrization Postulate (for indistinguishable particles): For identical particles that are indistinguishable, the total wave function describing the system must be either symmetric or antisymmetric under particle exchange. This postulate reflects the fundamental property of quantum mechanics that the exchange of identical particles should not result in any observable change. For particles that obey the symmetric wave function, they are called bosons, while particles that obey the antisymmetric wave function are called fermions.
Occupation Number Postulate: This postulate states that each quantum state in a many-particle system can be occupied by zero, one, or multiple particles. The occupation of each state is characterized by an occupation number, which indicates the number of particles in that state. For fermions, the occupation number can only be 0 or 1 due to the Pauli exclusion principle, which states that no two identical fermions can occupy the same quantum state. For bosons, the occupation number can be any non-negative integer.
Statistical Operator Postulate: The statistical operator, also known as the density operator, is used to describe the statistical properties of a quantum system. It is defined as a positive semi-definite operator that represents the density matrix of the system. The statistical operator captures the probabilities and correlations between different quantum states of the system. It allows for the calculation of various statistical quantities such as the expectation values of observables and the computation of probabilities for different measurement outcomes.
These postulates form the foundation of quantum statistics and are used to describe the behavior and properties of ensembles of identical particles in quantum mechanics. They provide a framework for understanding and predicting statistical distributions, energy levels, and other relevant quantities in systems consisting of indistinguishable particles.