The JWKB approximation follows from the assumption of an exponential form for the configuration-space wavefunction

[1, 2]. To first order in a small parameter

, the approximation works out as an integral over recognizable classical variables,

,

with domain

. Harmonic oscillation involves a purely quadratic Hamiltonian constraint,

. The energy variable

is quantized according to a stationary phase condition,

,

where offset

follows from the accumulation of phase shifts

at both turning points [1]. A wavefunction

is associated with each quantized energy

. Complete specification of each

requires superposition of left and right moving sinusoidal waves in the classically allowed region, where

, and attaches exponentially decaying evanescent solutions in the classically forbidden regions, where

. To construct such a solution in practice, we need to develop more complex geometry.

Identifying complex coordinates,

, allows for construction in the four-dimensional space spanned by

. Here the quadratic Hamiltonian determines a two-dimensional Riemannian surface, the

*harmonic hyperboloid*,

,

with a time-parametrization given as

.

Complex time,

, is said to provide a

*uniform domain* for the surface

. Trigonometric functions along real-valued coordinate

repeat after an interval of

, while the hyperbolic functions along real-valued coordinate

never repeat. Single periodicity clearly identifies the genus

. For an alternative proof, we introduce a new coordinate

and write

,

which only loses a phase degree of freedom relative to four-dimensional surface

. Level sets along the

axis are circles of increasing radius, and these are in bijection with the level sets of a projective sphere. For describing surfaces

and

, the adjective "Kleinian" may be more apt, as Riemann reportedly did not think in terms of uniform domains [3].

Back to quantum mechanics, the JWKB approximation imposes a restriction

or

, which limits a single integration contour between

and

,

.

Unfortunately, integration along this contour goes through a branch point where

, and the approximation necessarily diverges as

. Instead we avoid the turning point with an ansatz,

,

which maps range

onto a finite domain

. Changing parameters, we may then calculate the wavefunction in parametric form,

,

,

,

,

with dependence of

upon

suppressed, and

or

for even or odd parity. In making this approximation, we simply ignore small, nonzero values of the imaginary part

and assume that

.

The ansatz parameters

explore a space of trial wavefunctions, which includes the JWKB solution in a limit where

. Decreasing

from

toward

relaxes the wavefunction and smooths out the divergences. Meanwhile parameter

limits deviation from JWKB to a window of finite width. In this Demonstration, optimal values for parameters

can be found by quick overlap comparison with exact wavefunctions. Although nonstandard, this approach yields good-enough agreement to be considered effective in its own right.

[1] E. J. Heller,

*The Semiclassical Way to Dynamics and Spectroscopy*, Princeton, NJ: Princeton University Press, 2018.

[2] M. S. Child,

*Semiclassical Mechanics with Molecular Applications*, New York: Oxford University Press, 1991.

[3] J. J. Gray,

*Linear Differential Equations and Group Theory from Riemann to Poincaré*, 2nd ed., Boston: Birkhäuser, 2008.