= Solution
Let the infinite outer walls be at $x=0,D$, the finite steps at $x=a,b$, and the three <potential energy> values be $V_L,V_B,V_R$. The original drawing specifies the ordering
$$
V_L<E_0<E_1<V_R<E_2<V_B.
$$
The <energy eigenstates> can be chosen real. The <Dirichlet boundary conditions> are $\psi(0)=\psi(D)=0$; <wavefunction matching at a finite potential step> requires both $\psi$ and $\psi'$ to be continuous at $a,b$. In particular, neither finite step acts as an additional hard wall.
The <Time-independent Schrödinger equation> gives
$$
\psi''=\frac{2m}{\hbar^2}(V-E)\psi.
$$
Thus a <wavefunction> is sinusoidal in a <classically allowed region>, with <wavenumber> $k=\sqrt{2m(E-V)}/\hbar$, and a <linear combination> of <exponential functions> in a <classically forbidden region>, with inverse decay length $q=\sqrt{2m(V-E)}/\hbar$. Its <curvature> changes sign relative to $\psi$ at a step crossing $E$; its <slope> does not jump.
By the <Sturm oscillation theorem>, the three <energy eigenstates> have exactly \b[zero, one, and two interior nodes], respectively. Choose their <global phases> so that their slopes at the left wall are positive:
* $E_0$: a positive lobe in the left-hand <potential well>, followed by a nonoscillatory tail through the barrier and right-hand region. The tail reaches zero at the right wall. There are no interior <nodes>.
* $E_1$: two lobes of opposite sign in the left-hand <potential well>, with one interior <node>, followed by a nonoscillatory tail of the sign of the second lobe. Both the middle and right regions are <classically forbidden regions>.
* $E_2$: sinusoidal behavior in both outer regions, joined through the nonoscillatory barrier. There are two interior <nodes> in total. The right-hand <wavelength> is longer than the left-hand <wavelength>, because $E_2-V_R<E_2-V_L$.
For $E_0,E_1$, the right-hand piece is proportional to $\sinh[q_R(D-x)]$, rather than a pure decaying exponential: this enforces the finite-distance hard wall exactly. The barrier pieces likewise contain both exponential solutions. Their small tails are an instance of <quantum tunnelling>.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2022/paper-1-a1-wavefunctions.png]
{title=Illustrative lowest three wavefunctions for the potential ordering in A1}
This is an original illustrative calculation, with vertical lines marking only the region boundaries. It uses $D=1$, $a=0.4$, $b=0.6$, $V_L=0$, $V_R=0.62$, $V_B=0.82$, and $\hbar^2/(2mD^2)=0.002$ in arbitrary <energy> units. Its lowest <energy eigenvalues> are approximately $0.098,0.384,0.696$, so it has exactly the required ordering. Here both <nodes> of the third <wavefunction> happen to lie towards the left; its dominant lobe is on the right. These precise positions and relative amplitudes are illustrative: the original drawing supplies no numerical lengths, barrier heights, or <mass>, so it does not determine a unique quantitative <wavefunction>. The required sketch features are the matching, <curvature>, hard-wall zeros, and ordered <node> counts.
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