= Solution
Choose the centre of the <finite square well> as the origin and its edges as $\pm a$. The first three bound <wavefunctions> have \b[even, odd, even <parity>], with respectively \b[zero, one and two interior <nodes>]. They oscillate inside and decay exponentially outside, with $\kappa_n=\sqrt{2m(U-E_n)}/\hbar$. Both the <wavefunction> and its first <derivative> are continuous at each finite potential step. The higher <bound states> have longer external decay <lengths> because they lie closer to the barrier height. Overall sign is arbitrary.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2014/paper-1-finite-well.png]
{title=Original normalized finite-well eigenfunctions with parity, nodes and exponential tails; illustrative dimensionless depth, not supplied numerical data}
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