What is criteria of wave function in quantum mechanics?

The wave function ψ must be continuous. All its partial derivatives must also be continuous (partial derivatives are ∂ψ/ ∂x, ∂ψ/∂yetc.). This makes the wave function 'smooth'.

The wave function ψ must be quadratically integrable. This means that the integral

∫ψ * ψ dτ must exist.

Since ∫ψ * ψ dτ is the probability density, it must be single valued. The wave functions must form an orthonormal set. This means that

the wave functions must be normalised.∞

∫ψ * ψ dτ = 1i

the wave functions must be orthogonal.

∫ψ * ψ dτ = 0

dτ = δij where δij = 1 when i = j and δij = 0 when i ≠ j

δij is called Kronecker delta The wave function must be finite everywhere.

The wave function must satisfy the boundary conditions of the quantum mechanical system it represents.