= Solution
Inside the <infinite square well>, the <time-independent Schrödinger equation> and <Dirichlet boundary conditions> are
$$
-\frac{\hbar^2}{2m}\psi''=E\psi\quad(0<x<a),\qquad \psi(0)=\psi(a)=0,
$$
and the <wavefunction> vanishes outside the interval. For a nonzero state, <integration by parts> gives
$$
E\int_0^a|\psi|^2\,dx=\frac{\hbar^2}{2m}\int_0^a|\psi'|^2\,dx>0.
$$
The strict inequality follows because a function with zero derivative and zero endpoint values is zero. Put $k=\sqrt{2mE}/\hbar$. The general solution is $A\sin kx+B\cos kx$; the first <boundary condition> gives $B=0$, and the second gives $ka=n\pi$, $n=1,2,\ldots$. Since $\int_0^a\sin^2(n\pi x/a)\,dx=a/2$, the normalized <energy eigenstates> and <eigenvalues> are
$$
\boxed{\psi_n(x)=\sqrt{\frac2a}\sin\frac{n\pi x}{a},\qquad E_n=\frac{n^2\pi^2\hbar^2}{2ma^2},\quad n\geq1.}
$$
The normalization fixes the magnitude of the coefficient; its arbitrary <global phase> has been chosen real and positive.
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