= Ground state at the top of a central rectangular barrier
{title2=$\psi_{\rm middle}=\text{constant}$}
Consider hard walls at $x=0,3L$, zero <potential energy> on $0<x<L$ and $2L<x<3L$, and a central rectangular barrier whose height equals the <ground state> energy $E$. The <Time-independent Schrödinger equation> gives $\psi''=0$ on the central interval. Reflection symmetry and the <nodeless theorem for a one-dimensional ground state> require an even, positive state about $3L/2$, hence a constant central segment. <Wavefunction matching at a finite potential step> makes the derivatives vanish at $L,2L$. The side segments are proportional to $\sin(\pi x/(2L))$ and $\sin(\pi(3L-x)/(2L))$. Their common plateau value is $1/\sqrt{2L}$ after <wavefunction normalization>, and $E=\pi^2\hbar^2/(8mL^2)$. Equality of barrier height and energy is essential: a higher barrier gives positive curvature of a positive wavefunction in the middle, while a lower barrier gives negative curvature.
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