The oscillator spectrum came from together with the fact that a squared length is…
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
The oscillator spectrum ℏω(n+21) came from [a^,a^†]=1 together with the fact that a squared length is never negative. Because a^† is the adjoint of a^, that fact makes ⟨ψ∣N^∣ψ⟩=∥a^∣ψ⟩∥2≥0 automatically, so it cannot be switched off while a^† keeps its meaning. To see what the commutator does on its own, replace a^† by an operator c^ with no adjoint relation to a^: let a^ and c^ act on a space of functions that carries no inner product, obey only [a^,c^]=1, and set N^=c^a^.
((a)) Show, using only the commutator, that if N^∣ν⟩=ν∣ν⟩ then a^∣ν⟩, whenever it is nonzero, is an eigenvector of N^ with eigenvalue ν−1, and confirm that no step uses a length or an inner product.
((b)) Take a^=d/dz and c^=z (multiplication by z), acting on functions of z>0. Check that [a^,c^]=1, show that zν is an eigenvector of N^ with eigenvalue ν for every real ν, and show that for ν=π the descending chain a^kzπ never reaches the zero function, so its eigenvalues are unbounded below.
((c)) State in one sentence which single assumption, restored, forces the descending chain to stop at eigenvalue 0 and so recovers the non-negative integer spectrum.