Someone works through the translation operator as follows."Since for every position…
PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · The formalism of quantum mechanics · The formalism of quantum mechanics
The Structure of Quantum Mechanics, Ch Position, momentum and the origin of the canonical commutator
Problem
Someone works through the translation operator as follows."Since T^(a)∣x⟩=∣x+a⟩ for every position eigenvector, and ∣ψ⟩=∫dxψ(x)∣x⟩, it follows that
T^(a)∣ψ⟩=∫dxψ(x)T^(a)∣x⟩=∫dxψ(x)∣x+a⟩.
Relabelling the dummy integration variable as x′=x+a, so that x=x′−a, this is
T^(a)∣ψ⟩=∫dx′ψ(x′−a)∣x′⟩,
so the new wave function seen at the point labelled x′ is ⟨x′∣T^(a)∣ψ⟩=ψ(x′−a). But since T^(a) shifts every basis label x up by a, the wave function itself must shift up by a too, so the final answer is
⟨x∣T^(a)∣ψ⟩=ψ(x+a)."
((a)) Find the step where the error occurs.
((b)) State the correct relation between ⟨x∣T^(a)∣ψ⟩ and ψ, and explain in one sentence why a positive a moves the graph of the wave function to the right rather than shifting its formula by +a.