# Overlap of Two Wave Functions

Image »Overlap of Two Wave Functions« download Sharing and adapting of the illustration is allowed with the reference "Alexander Fufaev (fufaev.org)"
One-dimensional real wave functions $$\class{red}{\mathit{\Psi}}$$ and $$\class{blue}{\mathit{\Phi}}$$ are shown. The overlap is measured using the inner product (or scalar product) $$\langle\class{blue}{\mathit{\Phi}} | \class{red}{\mathit{\Psi}} \rangle$$:
• If the inner product is $$\langle\class{blue}{\mathit{\Phi}} | \class{red}{\mathit{\Psi}} \rangle = 1$$, then the corresponding wave functions $$\class{blue}{\mathit{\Phi}}$$ and $$\class{red}{\mathit{\Psi}}$$ are exactly on top of each other.
• If the inner product is $$\langle\class{blue}{\mathit{\Phi}} | \class{red}{\mathit{\Psi}} \rangle = 0$$, then the wave functions $$\class{blue}{\mathit{\Phi}}$$ and $$\class{red}{\mathit{\Psi}}$$ do not overlap at all.
• All values of the inner product $$\langle\class{blue}{\mathit{\Phi}} | \class{red}{\mathit{\Psi}} \rangle$$ between 1 and 0 result in a partial overlap of the two states.

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