Advanced Quantum Mechanics Schwabl · Problem 1.3

Second Quantization

Physics — Quantum Mechanics · bosons-creation-annihilation-operators

Problem

Verify the Bose commutation relations

[a_i, a_j] = 0, \quad [a_i^\dagger, a_j^\dagger] = 0, \quad [a_i, a_j^\dagger] = \delta_{ij}

by acting on an arbitrary state

|\ldots, n_i, \ldots, n_j, \ldots\rangle

and using the definitions of the creation and annihilation operators (1.3.4) and (1.3.6).