= Solution
The position <probability> <density> is
$$
p(x)=\frac2L\sin^2\frac{2\pi x}{L}\quad(0\leq x\leq L),
$$
and zero outside. It has zeros at $0,L/2,L$ and equal maxima $2/L$ at $L/4,3L/4$. The two lobes have equal integrated <probability>. The vertical axis is a <density> with units of inverse length, not a dimensionless <probability> at a point.
\Image[/past-exam-of-the-physics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1-probability-wells.png]
{title=Second-state position densities for an infinite well and an illustrative finite well, including evanescent tails}
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