PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Wave mechanics in one dimension
The Structure of Quantum Mechanics, Ch The oscillator as an algebra
Problem
Within the ladder algebra [a^,a^†]=1, suppose the number states {∣n⟩:n=0,1,2,…}, generated by repeated action of a^† on a normalised solution ∣0⟩ of a^∣0⟩=0, form a complete orthonormal basis of the oscillator's state space, so any state ∣χ⟩ can be written ∣χ⟩=∑n=0∞cn∣n⟩.
(a) Suppose ∣0′⟩ is another normalised state with a^∣0′⟩=0. Expand it in the number-state basis and use a^∣n⟩=n∣n−1⟩ to show that every coefficient cn with n≥1 must vanish.
(b) Conclude what ∣0′⟩ must be, given that both ∣0′⟩ and ∣0⟩ are normalised, and say in what sense the ground state is unique.
(c) Name, in one sentence, the assumption about the realisation that the chapter used to prove the completeness assumed at the outset, and say why that assumption cannot be dropped for an arbitrary realisation of [a^,a^†]=1.