Q1(k). Conditions for an acceptable wave function

Answer in brief

ψ must be single-valued, finite, continuous with a continuous first derivative, and square-integrable.

Key points

• Single-valued: the probability density ψ*ψ at a point must be unique.
• Finite everywhere: a probability cannot be infinite.
• Continuous, with a continuous first derivative: d²ψ/dx² in the Schrödinger equation must be defined (except where V jumps to infinity).
• Square-integrable, ψ → 0 at infinity: ∫ψ*ψ dτ must be finite so that ψ can be normalised to 1.