Phase space optimization of quantum representations: Direct-product basis sets

Abstract

The quantitative phase space similarities between the uniformly mixed ensembles of eigenstates, and the quasiclassical Thomas–Fermi distribution, are exploited in order to generate a nearly optimal basis representation for an arbitrary quantum system. An exact quantum optimization functional is provided, and the minimum of the corresponding quasiclassical functional is proposed as an excellent approximation in the limit of large basis size. In particular, we derive a stationarity condition for the quasiclassical solution under the constraint of strong separability. The corresponding quantum result is the phase space optimized direct-product basis—customized with respect to the Hamiltonian itself, as well as the maximum energy of interest. For numerical implementations, an iterative, self-consistent-field-like algorithm based on optimal separable basis theory is suggested, typically requiring only a few reduced-dimensional integrals of the potential. Results are obtained for a coupled oscillator system, and also for the 2D Henon–Heiles system. In the latter case, a phase space optimized discrete variable representation (DVR) is used to calculate energy eigenvalues. Errors are reduced by several orders of magnitude, in comparison with an optimized sinc-function DVR of comparable size.