= Wavefunction normalization
{title2=$\int|\psi|^2\,dx=1$}
Wavefunction normalization makes the total <Born rule> probability equal to one. In an <orthonormal basis> of <energy eigenstates>, $\psi=\sum_n c_n\phi_n$ is normalized precisely when $\sum_n|c_n|^2=1$.
For a proposed <infinite square well> mode $A\sin(n\pi x/L)$, the <integral> of its squared <modulus> is $|A|^2L/2$. Thus <wavefunction normalization> requires $|A|=\sqrt{2/L}$; choosing $A$ real and positive fixes a convenient <global phase>. For the equal <quantum superposition> $(\phi_1+\phi_2)/\sqrt2$, <orthogonality> makes the integrated cross term zero, leaving $1/2+1/2=1$. The cross term can nevertheless be nonzero locally: normalization constrains total probability, not the shape of its <probability density>.
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