Someone argues: "The number operator has eigenvalues spaced equally by
PHYSICS · Subjective · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch The oscillator as an algebra
Problem
Someone argues: "The number operator N^=a^†a^ has eigenvalues 0,1,2,… spaced equally by 1. An equally spaced spectrum is automatically non-degenerate, because if two orthogonal states shared an eigenvalue the whole ladder built on each of them would double up every level, which is impossible for a spectrum that is equally spaced. In particular the eigenspace for ν=0 is one-dimensional, so the state annihilated by a^ is unique."Find the error in this argument, and state in one or two sentences what actually establishes that the ground state is unique.