= Solution
{c}
The one-dimensional <time-independent Schrödinger equation> is
$$
-\frac{\hbar^2}{2m}\psi''(x)+V(x)\psi(x)=E\psi(x).
$$
For 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 $\psi_n''=-(n\pi/a)^2\psi_n$. Substitution gives the <energy levels>
$$
\boxed{E_n=\frac{n^2\pi^2\hbar^2}{2ma^2}.}
$$
Normalization requires $1=\int_0^a|\psi_n|^2dx=A^2a/2$. The requested real positive constant is therefore
$$
\boxed{A=\sqrt{\frac2a}.}
$$
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