= Solution
Inside the <infinite square well>, the time-independent <Schrödinger equation> is $-(\hbar^2/(2m))\psi''=E\psi$. The infinite exterior potential imposes $\psi(0)=\psi(a)=0$. For $E>0$, the first condition leaves a sine and the second requires $ka=n\pi$. A zero or negative energy gives only the zero solution under these conditions. The normalized <energy eigenstates> and <eigenvalues> are
$$
\boxed{\psi_n(x)=\sqrt{\frac2a}\sin\frac{n\pi x}{a},\qquad E_n=\frac{\hbar^2\pi^2n^2}{2ma^2},\quad n=1,2,\ldots.}
$$
The <wavefunctions> vanish outside the well, and a constant phase is immaterial. Normalization uses $\int_0^a\sin^2(n\pi x/a)\,dx=a/2$.
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