An approximation and alternative-formalism concept defining methods for computing quantum predictions when exact solutions are impractical. It governs controlled expansions, action-based formulations, and phase-space representations that support analytic and numerical work. It does not ensure accuracy outside its regime of validity and requires explicit error assessment or convergence checks. It enables tractable estimates of spectra, transition rates, and dynamical behavior across a wide range of models. The concept is generally stable, though improved algorithms and convergence techniques evolve over time.

Principle

Impose asymptotic matching of WKB expansions and conservation of probability/current so that exponentially growing, nonnormalizable components are excluded and phase-connection rules (Stokes phenomena) determine relative amplitudes.

Demonstration

For a one-dimensional potential with a single turning point, require that the WKB solution in the classically forbidden region decays exponentially as x → +∞; match this decaying solution through the turning point to oscillatory WKB solutions on the allowed side using the standard connection formulas to determine amplitude and phase shifts.

Misapplication

Forcing an arbitrary algebraic boundary condition (e.g., setting the wavefunction or derivative to zero at an internal point without physical justification) can pick the wrong WKB branch, produce spurious quantization, or violate current conservation across turning points.

Consequence

Correct boundary conditions ensure normalizable semiclassical eigenfunctions, consistent connection across turning points, and reliable semiclassical quantization rules such as Bohr–Sommerfeld conditions with the proper Maslov phase corrections.

Reversal

If one instead allows both exponentially growing and decaying solutions at an open boundary, the semiclassical approximation yields nonphysical, nonnormalizable states and ambiguous amplitude ratios; reversing the sign choice of Stokes multipliers changes interference and predicted tunneling rates.

Boundary

Applies to semiclassical approximations where the potential varies slowly on the de Broglie wavelength scale except near isolated turning points; excludes problems where WKB fails completely (e.g., strongly singular potentials, dense turning-point accumulation, or regions requiring uniform approximations).

Semantic Tension

Tension exists between local matching at a turning point (microlocal connection rules) and global boundary conditions (quantization on a finite interval): local WKB connection may allow multiple continuations, while global normalization picks a unique physical solution.

Synthesis

The WKB solution boundary condition is the rule that selects decaying versus growing semiclassical branches and fixes their relative phases and amplitudes by matching local Stokes-connection formulas with global normalization or physical open-boundary behavior, thereby producing physically acceptable semiclassical wavefunctions.