Zahloperator

Zahloperator, commonly referred to in English as the number operator, is a fundamental operator in quantum mechanics that counts the number of quanta in a given mode of a quantized field or harmonic oscillator. Its eigenvalues are nonnegative integers, corresponding to the occupation number of the mode.

In a single bosonic mode with creation and annihilation operators a† and a satisfying [a, a†] = 1, the number operator is defined as N = a† a. It acts on the Fock (number) basis |n> as N|n> = n|n>, where n = 0, 1, 2, …. The operators obey [N, a] = -a and [N, a†] = a†, which reflects how the ladder operators change the occupation number.

For the quantum harmonic oscillator, the Hamiltonian is H = ħω(N + 1/2). Thus, each eigenstate |n> has energy E_n = ħω(n + 1/2). In multi-mode systems, the total Hamiltonian is a sum over modes, H = Σ_k ħω_k (N_k + 1/2), with N_k counting excitations in mode k.

Applications of the Zahloperator include the description of photon-number statistics in quantum optics, analysis of coherent and thermal states, and formulations of second quantization in quantum field theory. In the Fock basis, the number operator is diagonal, providing a natural observable for counting quanta in a mode and enabling straightforward calculation of probability distributions for measurement outcomes.

See also: creation and annihilation operators, Fock space, quantum harmonic oscillator, second quantization, photon counting.