PHYSICS · Multi-part · 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: "Because [a^,a^†]=1 gives a^a^†=a^†a^+1, the order in which a^ and a^† are written in the Hamiltonian is a free relabelling with no physical content. So if I define H^′=ℏωa^a^† instead of the usual H^=ℏω(a^†a^+21), I am just choosing where the zero of energy sits, and the ground-state energy of H^′ must come out as −ℏω/2, the mirror image of the usual +ℏω/2."
((a)) Using only [a^,a^†]=1 and boundedness of the spectrum of H^′ below, find the actual spectrum of H^′=ℏωa^a^†.
((b)) For ω=3.5×1013rads−1, give the numerical value in eV of the ground-state energy of H^′, and say in one sentence why the quoted argument is wrong.