Position matrix elements in a symmetric infinite square well ($⟨{\varphi }_k|x|{\varphi }_n⟩$)

For a well $-a<x<a$, choose normalized energy eigenstates ${\varphi }_n=a^{-1/2}\sin [(n+1)\pi (x+a)/(2a)]$, $n=0,1,\dots$. The diagonal position matrix elements vanish by the parity operator. For $k\ne n$,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, whileTaking 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.