= Solution
The <Sturm oscillation theorem> gives \b[three <nodes of a wavefunction>] for the fourth <bound state>. <reflection symmetry> and <nondegeneracy of one-dimensional bound states> make it \b[odd]: one <node of a wavefunction> is at the centre, and the other two lie at opposite positions. In each flat allowed region the <wavefunction> is a <sinusoid>, with $k=\sqrt{2m(E-V)}/\hbar$; its <wavelength> is shortest in the central, deepest region. Outside the well it decays exponentially, with opposite signs on the two sides. Both $\Psi$ and $\Psi'$ remain continuous at every finite step, as required by <wavefunction matching at a finite potential step>; only the curvature jumps. In particular, the <wavefunction> need not vanish at a step.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2011/paper-2-stepped-state.png]
{title=Fourth bound state}
This original <wavefunction matching at a finite potential step> illustration uses freely chosen well dimensions and <energies>. It shows the required <parity>, node count, <wavelength> ordering and decaying tails; the source drawing supplies no numerical scale from which to fix exact node positions.
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