For the radial equation

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Central potentials and canonical three-dimensional systems · Central potentials and canonical three-dimensional systems

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

For the

l=0

radial equation

-\dfrac{\hbar^2}{2\mu}u''(r) + V(r)u(r) = Eu(r)

, with

V(r)

finite everywhere (including at

r=0

), someone argues:"The equation is second order, so near

r=0

it has two independent solutions: one with

u(0)=0

, and one with

u(0)\ne 0

(equivalently

R(r)=u(r)/r \to \text{const}/r

r\to 0

). Normalisability is the only thing that can rule a solution out, and

\int_0^\infty |R(r)|^2 r^2\,dr = \int_0^\infty |u(r)|^2\,dr

is perfectly finite for the

u(0)\ne 0

solution too, because the factor

r^2

in the measure exactly cancels the

1/r^2

|R|^2

near the origin, leaving no divergence there. So both solutions are equally acceptable on normalisability grounds, and setting

u(0)=0

is just a convention." ((a)) Is the claim that

\int_0^\infty|u|^2 dr

stays finite for the

u(0)\ne 0

solution correct? ((b)) Find the actual error in the argument, and state the correct reason

u(0)=0

is required.

Related theory

Quantum Mechanics

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