What mathematical conditions must a wave function satisfy to be physically acceptable?

In one line

A physically acceptable (well-behaved) wave function must be single-valued, finite, continuous, have a continuous first derivative, and be normalisable so the total probability is 1.

Explained simply

The wave function gives a probability, and probabilities must make sense. At any place, the chance of finding a particle cannot have two values, cannot be infinite and cannot jump suddenly. These common-sense requirements become the mathematical conditions on \(\psi\).

How it works

1. Single-valued: at each point \(\psi\) has only one value, otherwise the probability there would be ambiguous.
2. Finite: \(\psi\) must not become infinite anywhere, since probability cannot be infinite.
3. Continuous: \(\psi\) must not jump suddenly between neighbouring points.
4. First derivative continuous: \(d\psi/dx\) must be continuous wherever the potential is finite, because Schrödinger's equation contains \(d^2\psi/dx^2\). (At an infinite potential wall, as in the rigid box, only \(\psi\) itself must be continuous.)
5. Normalisable (square integrable): \(\int_{-\infty}^{\infty}|\psi|^2dx\) must be finite so it can be made equal to 1. Hence \(\psi \to 0\) as \(x \to \pm\infty\).

These conditions, applied as boundary conditions, are what give rise to quantisation: only certain energies allow a well-behaved solution.

Key formula

\(\int_{-\infty}^{\infty}|\psi|^2dx = 1\) (normalisation); \(\psi\) and \(d\psi/dx\) continuous at boundaries where \(V\) is finite.

Quick example

Check \(\psi = e^{-|x|}\): single-valued, finite, continuous, and \(\int|\psi|^2dx = 1\) is finite, so it is normalisable. But its slope jumps from +1 to −1 at \(x = 0\), so it is acceptable only where the potential has a special (delta-like) spike there. \(\psi = e^{x}\) fails because it becomes infinite as \(x \to \infty\). \(\psi = \tan x\) fails because it is infinite at \(x = \pi/2\).

Common mistakes

• Forgetting normalisability, the most commonly missed condition.
• Saying \(\psi\) must be real; it can be complex.
• Insisting that \(d\psi/dx\) be continuous at an infinite potential wall.

How MU asks it

Asked as "State the conditions for a well-behaved wave function" or "Write the mathematical conditions on a wave function" (3 marks). List the conditions with a one-line reason each; adding an example of an unacceptable function strengthens the answer.