Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / J / 1 / Solution

The one-dimensional time-independent Schrödinger equation isFor the infinite square well on $[0,a]$, $V=0$ inside and the wave function is zero outside. The boundary conditions are $\psi (0)=\psi (a)=0$. For positive integer $n$, ${\psi }_n=A\sin (n\pi x/a)$ satisfies them, and ${\psi }_n^{\prime \prime }=-(n\pi /a{)}^2{\psi }_n$. Substitution gives the energy levelsNormalization requires $1={\int }_0^a|{\psi }_n{|}^{2}dx=A^{2}a/2$. The requested real positive constant is therefore