Principles of Quantum Mechanics · Problem 12.6.4

Rotational Invariance and Angular Momentum — R. Shankar

Physics — Quantum Mechanics · delta-function-laplacian

Problem

(1) Show that δ3(r−r′)≡δ(x−x′)δ(y−y′)δ(z−z′)=1r2sin⁡θ δ(r−r′)δ(θ−θ′)δ(ϕ−ϕ′)\delta^3(\mathbf{r} - \mathbf{r}') \equiv \delta(x-x')\delta(y-y')\delta(z-z') = \frac{1}{r^2\sin\theta}\,\delta(r-r')\delta(\theta-\theta')\delta(\phi-\phi') (consider a test function). (2) Show that ∇2(1/r)=−4πδ3(r)\nabla^2(1/r) = -4\pi\delta^3(\mathbf{r}) (Hint: First show that ∇2(1/r)=0\nabla^2(1/r) = 0 if r≠0r \neq 0. To see what happens at r=0r = 0, consider a sphere centered at the origin and use Gauss's law and the identity ∇2ϕ=∇⋅∇ϕ\nabla^2\phi = \mathbf{\nabla}\cdot\mathbf{\nabla}\phi.)

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