Choose zero potential for . In an infinite square well,
(1)
The first three levels have ratios , with zero, one and two interior nodes; the wavefunctions vanish outside.
For a finite square well with exterior potential , bound states have , sinusoidal interiors and tails with . Matching wavefunction and derivative for equal masses gives
(2)
when parity is taken about the well center. Corresponding energies fall below the infinite-barrier values, and the wavefunctions spread beyond the walls. There are finitely many bound states; a shallow well need not possess three. States above form a scattering continuum.
Figure 1.
Wavefunctions, probability densities and core-shell confinement
. Original normalized infinite- and finite-well calculations with a chosen dimensionless barrier, plus qualitative type-I envelope sketches. The envelope panel is not a two-particle probability density or material-specific fit. Details.

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