the finite square well
Take the particle-in-a-box and make its walls a finite height instead of infinite -- a rectangular dip of depth V_0 and width L in the potential. Now the trap is escapable in principle, and the physics becomes richer and more honest: there are only finitely many bound states, their wavefunctions do not slam to zero at the edges but leak into the walls, and whether a given level exists at all depends on how deep and wide the well is. This is the model that teaches you what a real bound state looks like.
Inside the well the Schrodinger equation gives oscillating sines and cosines just as before, with wavenumber k = sqrt(2 m E)/hbar; but outside, where E < V_0, the solution is a decaying exponential exp(-kappa |x|) with kappa = sqrt(2 m (V_0 - E))/hbar. Requiring the wavefunction and its slope to be continuous at each wall gives transcendental matching conditions -- for the symmetric (even) states, k tan(k L/2) = kappa, and for the antisymmetric (odd) states, -k cot(k L/2) = kappa -- which you solve graphically or numerically rather than in closed form. The number of bound states is set by the dimensionless well strength; roughly, a new level appears each time sqrt(2 m V_0) L / (pi hbar) crosses another integer.
The signature feature is that the wavefunction penetrates the classically forbidden region: outside the well, where a classical particle of that energy could never be, the quantum probability density is small but nonzero, decaying over a length 1/kappa. That evanescent tail is the seed of quantum tunnelling. A one-dimensional finite well always holds at least one bound state no matter how shallow, but -- an honest caveat -- this guarantee is special to one dimension; in three dimensions a well that is too weak binds nothing at all.
A neutron of energy E half the depth of a well of width L = 3 fm and depth V_0 has an exterior decay length 1/kappa = hbar / sqrt(2 m (V_0 - E)); for typical nuclear numbers this is a fraction of a femtometre, so the neutron's wavefunction pokes only a little way out of the nuclear well -- yet that little tail is what makes alpha and neutron emission possible.
The exponential leak of a bound state into a finite wall is the physical root of tunnelling and decay.
A common error is to force the wavefunction to zero at the walls as in the infinite box; here it must instead match smoothly onto a decaying exponential outside. And the 'at least one bound state' theorem holds in 1D only -- do not carry it into 3D, where a shallow well can bind nothing.