= Position matrix elements in a symmetric infinite square well
{title2=$\langle\phi_k|x|\phi_n\rangle$}
For a well $-a<x<a$, choose normalized <energy eigenstates> $\phi_n=a^{-1/2}\sin[(n+1)\pi(x+a)/(2a)]$, $n=0,1,\ldots$. The diagonal position <matrix elements> vanish by the <parity operator>. For $k\ne n$,
$$
\langle\phi_k|x|\phi_n\rangle=
\frac{8a(k+1)(n+1)[(-1)^{k+n}-1]}{\pi^2[(k+1)^2-(n+1)^2]^2}.
$$
To derive it, set $y=x+a$ and use $2\sin Ay\sin By=\cos[(A-B)y]-\cos[(A+B)y]$. For unequal positive integer indices $r,s$, the constant part of $x=y-a$ integrates to zero by <orthogonality>, while
$$
\int_0^{2a}(y-a)\cos\left(\frac{j\pi y}{2a}\right)dy
=\frac{4a^2[(-1)^j-1]}{j^2\pi^2}\quad(j\ne0).
$$
Taking the difference for $j=r-s$ and $j=r+s$ yields the formula after setting $r=k+1,s=n+1$. It vanishes for equal parity and gives $x_{01}=-32a/(9\pi^2)$ for this phase convention.
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