In applying the Numerov method to count the nodes in r, we assumed that changes sign as it goes through a node. However, there

In applying the Numerov method to count the nodes in ψr, we assumed that ψ changes sign as it goes through a node. However, there are functions that do not have opposite signs on each side of a node. For example, the functions y = x2 and y = x4 are positive on both sides of the node at x = 0. For a function y that is positive at points just to the left of x = a, is zero at x = a, and is positive just to the right of x = a, the definition y' = lim Δx→0 Δy / Δx shows that the derivative y' is negative just to the left of x = a and is positive just to the right of x = a. Therefore (assuming y' is a continuous function), y' is zero at x = a. (An exception is a function such as the V-shaped function y =  x, whose derivative is discontinuous at x = a. But such a function is ruled out by the requirement that ψ be continuous.)
(a) Use the Schrödinger equation to show that if ψ(x) = 0 at x = a, then ψ = 0 at x = a (provided V(a) ≠ ∞).
(b) Differentiate the Schrödinger equation to show that if both ψ and ψ are zero at x = a, then ψ"'(a) = 0 [provided V' (a) ≠ ∞]. Then show that all higher derivatives of ψ are zero at x = a if both ψ and ψ are zero at x = a (and no derivatives of V are infinite at x = a). If ψ and all its derivatives are zero at x = a, the Taylor series (4.85) shows that ψ is zero everywhere. But a zero function is not allowed as a wave function. Therefore, ψ and ψ cannot both be zero at a point, and the wave function must have opposite signs on the two sides of a node.