Riccati differential equation for quantum mechanical bound states: Comparison of numerical integrators

Abstract

Computational comparison is presented of several integrators for theRiccati differential equation for the one-dimensional quantum mechanical bound state problem. The computational method for the quantum Hamilton-Jacobi equation for stationary states is briefly reviewed. An interpolation formula for the quantum momentum function is proposed for the evaluation of the phase integral. The Möbius integrator, the R-matrix propagation method, the log derivative method, and the symplectic integrator can accurately pass through singularities in the solution. These four integrators are actually Möbius transformations with different coefficients. Analytical analysis and numerical results demonstrate that the classical momentum is a good choice for the initial condition to obtain the bound state solution of the Riccati differential equation as long as the starting point of the propagation is far enough from the origin. The computational comparison is demonstrated for the harmonic oscillator. The numerical results indicate that the symplectic integrators achieve highest accuracy, and the simplicity of these algorithms shows an advantage over the other methods.

Density dynamics in some quantum systems

“…[42][43][44][45] For stationary states, QHJE is solved to synthesize the wave function at the same time the reflection and transmission coefficients are calculated for several problems. [46,47] Quantum interference effects are demonstrated within the Bohmian formalism. [48,49] Bohm's analytic route is applied to solve problems related to surface science.…”

Applied Bohmian Mechanics

“…In other words, this complexified Bohmian mechanics is also an alternative synthetic method but in complex space [193]. With the same spirit and by the same time, an alternative synthetic approach was also developed by Chou and Wyatt [407][408][409], with applications to both bound states and scattering systems.…”

Grassmannian spectral shooting

“…The numerical construction of pure-point spectra is important in determining the linear stability of coherent structures. Examples of such structures are: ground and higher excited states of molecules in quantum chemistry (Johnson [56], Hutson [51], Gray and Manopoulous [38], Manopoulous and Gray [69], Chou and Wyatt [23,24], Ledoux [64], Ledoux, Van Daele and Vanden Berghe [65], Ixaru [55]); nonlinear travelling fronts in reaction-diffusion such as autocatalysis or combustion (Billingham and Needham [9], Metcalf, Merkin and Scott [72], Doelman, Gardner and Kaper [31], Terman [96], Gubernov, Mercer, Sidhu and Weber [41]); nerve impulses (Alexander, Gardner and Jones [2]); neural waves (Coombes and Owen [25]); solitary waves or steady flows over compliant surfaces (Pego and Weinstein [82], Alexander and Sachs [3], Chang, Demekhin and Kopelevich [21], Kapitula and Sandstede [57], Bridges, Derks and Gottwald [14], Allen [4], Allen and Bridges [5]); laser pulses (Swinton and Elgin [95]); nonlinear waves along elastic rods (Lafortune and Lega [62]); ionization fronts (Derks, Ebert and Meulenbroek [28]) or spiral waves (Sandstede and Scheel [89]). …”

“…First, evolving the solution subspaces, considered as curves in the Grassmann manifold is not new for autonomous problems; see Hermann and Martin [44,45,46,47,48], Martin and Hermann [71], Brockett and Byrnes [18], Shayman [92], Rosenthal [87], Ravi, Rosenthal and Wang [86], Zelikin [100], Abou-Kandil, Freiling, Ionescu and Jank [1] and Bittanti, Laub and Willems [12]. Using Riccati systems to solve nonautonomous spectral problems is also not new; see Johnson [56], Hutson [51], Pryce [85], Manopoulous and Gray [69], Gray and Manopoulous [38] and Chou and Wyatt [23,24]. Here Riccati systems correspond to the flow of the linear spectral problem projected onto the Grassmann manifold with a fixed coordinate patch representation; see Schneider [91].…”

“…Here Riccati systems correspond to the flow of the linear spectral problem projected onto the Grassmann manifold with a fixed coordinate patch representation; see Schneider [91]. Also see Schiff and Shnider [90] and Chou and Wyatt [24] who use this connection to integrate Riccati systems through singularities.…”