Matching a finite square well

Take $V(x)=-V_0$ for $|x|<a$ and zero outside, with $V_0>0$ and half-width $a$. Bound energies satisfy $-V_0<E<0$. Inside, kinetic energy $E+V_0$ is positive and solutions oscillate. Outside, negative total energy gives exponentials, of which only the decaying branches are square-integrable. A positive exterior energy instead permits scattering waves; it is not another member of the discrete bound spectrum.

At a finite jump of a potential with constant particle mass, both $\psi$ and $\psi'$ are continuous. Integrating Schrödinger's equation across the step shows that a finite potential contributes no finite derivative jump. A discontinuity in the wavefunction would also generate an unsupported singular second derivative. These conditions differ from a delta interaction and from an infinite-wall boundary. Writing the same second-order equation in each region is insufficient until the correct interface conditions are imposed.

The potential is reflection symmetric. Its ordinary one-dimensional bound levels are nondegenerate, so eigenfunctions can be chosen even or odd. Use $A\cos(kx)$ for an even interior state and $A\sin(kx)$ for an odd one. At the positive edge, continuity fixes the tail amplitude. Dividing the derivative condition by the value then removes the normalization constant and leaves a transcendental equation for energy. Reflection supplies the negative-edge condition automatically.

Dimensionless variables organize the roots without hiding the physical threshold. Set $z=ka$, $y=\kappa a$, and $z_0=a\sqrt{2mV_0}/\hbar$. Allowed roots lie on the circle $z^2+y^2=z_0^2$ with $y>0$. Even and odd matching curves intersect different branches. A pole of tangent is not an eigenvalue just because a numerical residual changes sign there; the continuous branch and the original matching conditions must both be checked.

For an even state the positive-side tail is $A\cos(ka)e^{-\kappa(x-a)}$. Integrating its square on both exterior half-lines gives $|A|^2\cos^2(ka)/\kappa$. The interior contribution is $|A|^2[a+\sin(2ka)/(2k)]$. Their sum, not the interior alone, must equal one. A shallow state can put substantial probability outside its nominal well; being bound means normalizable localization, not absolute exclusion from every region above the energy.

Deepening the well approaches infinite-well energies when measured above the floor, while exterior tails shrink. Near zero binding, the decay length $1/\kappa$ grows and a spatially large integration box is needed to normalize the state numerically. Matching and normalization answer different questions: a guessed energy may produce an easily normalized piecewise function that still fails derivative continuity. Conversely, an eigenfunction's overall amplitude is fixed only after its energy and matching ratios have been found.