Semiclassical Approximation for Quantum Harmonic Oscillator Wavefunctions
In semiclassical mechanics, wavefunction shapes are not entirely easy to determine; however, the Jeffreys–Wentzel–Kramers–Brillouin (JWKB) approximation gives an explicit and intuitive prescription [1, 2]. Testing on the quantum harmonic oscillator shows that a typical JWKB approximation accurately recreates the exact solution, except in the immediate vicinity of the turning points. To attain 99.8% or 99.9% agreement on the entire domain, we advance a two-parameter ansatz that relaxes the integration contour along the surface of a Riemannian harmonic hyperboloid (see Details).
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 . 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 .
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.
 E. J. Heller, The Semiclassical Way to Dynamics and Spectroscopy, Princeton, NJ: Princeton University Press, 2018.
 M. S. Child, Semiclassical Mechanics with Molecular Applications, New York: Oxford University Press, 1991.
 J. J. Gray, Linear Differential Equations and Group Theory from Riemann to Poincaré, 2nd ed., Boston: Birkhäuser, 2008.