Particlenumber

Particlenumber, or particle number, is the count of discrete constituents in a physical system. It is a central quantity in physics because it determines many thermodynamic, statistical, and dynamical properties. In nonrelativistic quantum systems and many-body physics, particle number can be well defined for a given set of modes or particle species. In relativistic quantum field theory, particle number is not always a conserved or globally meaningful quantity due to particle creation and annihilation processes, except in sectors where a corresponding conservation law applies.

In the formalism of second quantization, a particle-number operator N is defined by N = sum_k a_k† a_k, where a_k† and a_k are creation and annihilation operators for mode k. The eigenstates |n1, n2, ...> have eigenvalues N = sum_k n_k, representing the total number of particles. If the Hamiltonian commutes with N, [N, H] = 0, particle number is conserved and remains constant in time for closed systems.

In statistical mechanics, whether particle number is fixed or fluctuates depends on the ensemble. In the canonical ensemble, N is fixed. In the grand canonical ensemble, N fluctuates with the chemical potential μ, and the average ⟨N⟩ is derived from the grand partition function Z_G(μ, T); fluctuations are given by Var(N) = ⟨N^2⟩ − ⟨N⟩^2, which relates to derivatives of ln Z_G with respect to μ. In quantum optics and ultracold-atom experiments, particle-number statistics can be studied directly, revealing Poissonian, sub-Poissonian, or super-Poissonian distributions depending on the state.

In practice, the concept of particlenumber helps describe and quantify systems ranging from photons and atoms to quasiparticles, and it plays a key role in analyzing measurements, fluctuations, and conservation laws across multiple areas of physics.