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).

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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 [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 ,
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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.
References
[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.
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