= Solution
The <Sturm oscillation theorem> requires respectively \b[zero, one and two <nodes of a wavefunction>]. Treating the vertical outer boundaries as hard walls, all three <wavefunctions> vanish there. The <potential energy> is asymmetric, so no even/odd <parity> should be imposed. At internal finite steps, both the <wavefunction> and its first <derivative> are continuous.
For $E_0$ and $E_1$, only the left well is classically allowed. Their <wavefunctions> oscillate there and continue as evanescent combinations through the barrier and right region, reaching zero at the far wall. The first is <nodeless>; the second has one node in the left well. For $E_2$ both wells are allowed, while the central barrier remains forbidden: the <wavefunction> oscillates in both wells, with an evanescent bridge and two <nodes of a wavefunction> overall. A node may lie inside a finite forbidden barrier when the matching coefficients have opposite signs. Precise node positions and relative amplitudes require dimensions and heights not specified by the qualitative drawing.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2010/paper-2-asymmetric-states.png]
{title=Three asymmetric-well states}
The original illustration solves an explicitly chosen model with the same ordering of allowed and forbidden regions. Its numerical parameters and detailed node positions are illustrative.
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