Suppose someone proposes to realise the ladder algebra with matrices: a matrix
PHYSICS · Multi-part · 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
Suppose someone proposes to realise the ladder algebra with matrices: a^ a d×d matrix, a^† its conjugate transpose, obeying [a^,a^†]=1, where 1 is the d×d identity.
((a)) Take the trace of both sides of the commutation relation and show that no finite d works.
((b)) Reach the same conclusion from the ladder instead. Starting from a normalised ∣0⟩ with a^∣0⟩=0, which the descent argument supplies, use a^†∣n⟩=n+1∣n+1⟩ to show that the states ∣0⟩,∣1⟩,∣2⟩,… are all nonzero and mutually orthogonal, and say why that is impossible in d dimensions.
((c)) On the number states themselves, a^a^† and a^†a^ are both diagonal. Write each of their traces as a sum over n, and explain why the argument of part (a) cannot be run there.