Advanced Quantum Mechanics Schwabl · Problem 1.10

Spin-1/2 Fermions

Physics — Quantum Mechanics · field-operator-particle-number

Problem

(a) Show, by verifying the relation n(x)∣ϕ⟩=δ(x−x′)∣ϕ⟩,n(\mathbf{x})|\phi\rangle = \delta(\mathbf{x}-\mathbf{x}')|\phi\rangle, that the state ∣ϕ⟩=ψ†(x′)∣0⟩|\phi\rangle = \psi^\dagger(\mathbf{x}')|0\rangle (∣0⟩=|0\rangle = vacuum state) describes a particle with the position x′\mathbf{x}'. (b) The operator for the total particle number reads: N^=∫d3x n(x).\hat{N} = \int d^3x\, n(\mathbf{x}). Show that for spinless particles [ψ(x),N^]=ψ(x).[\psi(\mathbf{x}), \hat{N}] = \psi(\mathbf{x}).

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