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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.
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