While we have not yet solved the Hydrogen atom problem two allowed solutions are

$[\backslash$psi $\_1($r$)=$N$\_1$ e$^-$r$/$a$\_0$; $\backslash$psi $\_2($r $\backslash$theta $)=$N$\_2$ r sin$\backslash$theta cos$$ e$^-$r$/2$ a$\_0]$

N$\_1$ and N$\_2$ are normalization constants and a$\_0=$ constant $($Bohr radius$).$ a$.$ Determine N$\_1$ and N$\_2$ by normalizing the functions. Note: d $\backslash$tau $=$r$^2$ d r sin$\backslash$theta d $\backslash$theta d $,0<=$ r $<=\backslash$infty $,0<=\backslash$theta $<=\backslash$pi $,0<=<=2\backslash$pi b$.$ Show $\backslash$psi $\_1$ and $\backslash$psi $\_2$ are orthogonal.