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Off hand I didn't know the answer so I poked around. I remembered something like this on my graduate exam so here is the short answer. I think this will give you what you want if not let me know.

From: http://academic.reed.edu/chemistry/alan/Research/Bond/OneE/BoxProp/box.html

Wavefunction overlap 1D particle in a box (the equation is given on the website)

Quoted from the website:

`"You can easily show that the overlap of different wavefunctions is
necessarily zero. Consider, for example, if ø _{m} and ø_{n} have
different symmetries their product will be antisymmetric and the integral will
vanish. Different arguments are required for wavefunctions of the same
symmetry type, but the result is the same. S = 0 if 'm' and 'n' are different.
`

`
We indicate the 'no overlap' property of wavefunctions by saying they are
'orthogonal.' Valid wavefunctions with different quantum numbers must
always be orthogonal.
`

`
Note that a wavefunction can never be orthogonal to itself. Setting m = n
makes the overlap integral identical to the normalization integral, i.e.,
Smm = 1."
`

Also some handy references:

__Physical Chemistry__, 7th Edition, Peter Atkins and Julio de Paula
Applied Mathematics for Physical Chemistry by James R. Barrante

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