Shooting and matching for bound eigenstates

For a one-dimensional real potential, the time-independent Schrödinger equation is second order. Fix one boundary condition and choose a nonzero starting slope or neighboring value to set an arbitrary amplitude scale. Linearity permits normalization afterward. If both the wavefunction and its derivative are set to zero at a regular initial point, the resulting identically zero solution supplies no eigenstate. Vary energy instead of mistaking a change of amplitude for a new bound level.

A finite-difference recurrence makes this logic explicit. With zero potential and homogeneous Dirichlet endpoints, the dimensionless trial energy determines each next sample from two previous samples. The final sample is a mismatch function. Its zeros select eigenvalues of that discrete boundary problem. For smooth continuum shooting, Runge-Kutta or Numerov integration has separate step error; the recurrence below is deliberately only the simplest centered finite stencil, with no claim of Numerov accuracy.

For a localized state on an unbounded line, the physical exterior solution decays. Outward integration can amplify a tiny growing exponential contamination and overwhelm that solution. Integrating inward from a sufficiently distant asymptotic boundary and matching to an interior solution can improve conditioning. One convenient mismatch compares logarithmic derivatives, eliminating the unknown amplitude factor when neither matched wavefunction vanishes.

At a node the logarithmic derivative has a pole, so a sign change across that pole is not automatically an energy root. A Wronskian or another nonsingular matching condition can avoid that particular division. Finite jumps of a finite potential require continuity of wavefunction and derivative, whereas an infinite hard wall or a delta potential has different matching rules. Using a remembered condition without checking the operator's domain can therefore manufacture or miss levels.

In regular one-dimensional bound-state problems, node counts and symmetry help distinguish neighboring eigenstates. A numerical search should preserve its intended branch as spacing, matching point and outer boundary are changed. A root finder can converge tightly to the wrong level, to a pole of the mismatch, or to the spectrum of an overly small box. Report which state was tracked and how its energy and normalized shape respond to independent controls.

The finite square well offers an exact matching benchmark. An even solution is a cosine inside and a decaying exponential outside. Equating their logarithmic derivatives at the positive edge gives k times tangent of k times the half-width equal to the exterior decay constant. This relation is not an empirical fitting rule: it follows directly from derivative continuity and exposes the sign of the exponentially decaying tail. Its singular tangent points still require the same bracketing care as other logarithmic mismatches.