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International Journal of Quantum Chemistry

Nonsingular constraints in time‐dependent variational principle for parametrized wave functions

Katsuhisa Ohta

Published Sep 20, 2012• 1 citation
Publisher

Abstract

Abstract In this work, we consider two conditions required for the nonsingularity of constraints in the time‐dependent variational principle (TDVP) for parametrized wave functions. One is the regularity condition which assures the static nonsingularity of the constraint surface. The other condition is the second‐class condition of constraints which assures the dynamic nonsingularity of the constraint surface with a symplectic metric. For analytic wave functions for complex TDVP‐parameters, the regularity and the second‐class conditions become equivalent. The second‐class condition for expectation values is reduced to the noncommutability of the corresponding quantum operators. The symplectic singularity of the equation of motion of TDVP is also shown to be a local breakdown of the second‐class condition in an extended canonical phase‐space. © 2012 Wiley Periodicals, Inc.

DOI
10.1002/qua.24325

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Abstract In this work, we consider two conditions required for the nonsingularity of constraints in the time‐dependent variational principle (TDVP) for parametrized wave functions. One is the regularity condition which assures the static nonsingularity of the constraint surface. The other condition is the second‐class condition of constraints which assures the dynamic nonsingularity of the constraint surface with a symplectic metric. For analytic wave functions for complex TDVP‐parameters, the regularity and the second‐class conditions become equivalent. The second‐class condition for expectation values is reduced to the noncommutability of the corresponding quantum operators. The symplectic singularity of the equation of motion of TDVP is also shown to be a local breakdown of the second‐class condition in an extended canonical phase‐space. © 2012 Wiley Periodicals, Inc.
The time-dependent variational principle in the stationary action principle form is formulated with constraint conditions for parametrized wave functions. The constraint conditions are classified into the first and the second classes as in Dirac's constrained classical mechanics based on the commutability of operators. If the local bases in the parametrized wave functions construct a complete basis set, the classification of constraints with the commutator becomes equivalent to that with the complex generalized Poisson brackets (CGPB). However, in approximate wave functions with limited variational parameters, the equivalence between the CGPB and the commutator of operators is lost. The first-class constraints and the constants of motion classified with the commutator should be reconsidered as the pseudo-second-class constraints in the approximate wave functions.
The sensitivity analysis for the time-dependent variational principle (TDVP) with constraints for parametrized wave functions [K. Ohta, Phys. Rev. A 70, 022503 (2004)] is investigated to assess the geodesic deviation caused by external parameters. The constraints for the sensitivity functions, inherited from the TDVP, are dealt with their consistency conditions. As an example of the sensitivity analysis, the geodesic deviation in the neighborhood of stationary states is investigated for general wave functions.
A modification of Frenkel's time-dependent variational principle (TDVP) determines the complex phase of the wave function: the real and imaginary parts give the phase prescribed by the Schrodinger equation and the correct normalization, respectively. They are given by the hermitized and anti-hermitized components of the frequency operator. The quantum-mechanical TDVP is transcribed into a classical action principle where the hermitized and anti-hermitized components of the frequency operator are shown to be related to the velocity potential and stream function, respectively, of the complex potential method of two-dimensional hydrodynamic flow. The product of metric coefficients that appears in the Poisson brackets is the derivative of the circulation; the conservation of circulation, angular momentum, and particle number are all equivalent.
Abstract Time-dependent variational principle (TDVP) provides powerful methods in solving the time-dependent Schröinger equation. As such Kan developed a TDVP (Kan 1981 Phys. Rev. A 24, 2831) and found that there is no Legendre transformation in quantum variational principle, suggesting that there is no place for the Maupertuis reduced action to appear in quantum dynamics. This claim is puzzling for the study of quantum–classical correspondence, since the Maupertuis least action principle practically sets the very basic foundation of classical mechanics. Zambrini showed within the theory of stochastic calculus of variations that the Maupertuis least action principle can lead to the Nelson stochastic quantization theory (Zambrini 1984 J. Math. Phys. 25, 1314). We here revisit the basic aspect of TDVP and reveal the hidden roles of Maupertuis-Hamilton least action in the Schrödinger wavepacket dynamics. On this basis we propose a dual least (stationary) action principle, which is composed of two variational functionals; one responsible for ‘energy related dynamics’ and the other for ‘dynamics of wave-flow’. The former is mainly a manifestation of particle nature in wave-particle duality, while the latter represents that of matter wave. It is also shown that by representing the TDVP in terms of these inseparably linked variational functionals the problem of singularity, which is inherent to the standard TDVPs, is resolved. The structure and properties of this TDVP are also discussed.
We investigate variational principles for the approximation of quantum dynamics that apply for approximation manifolds that do not have complex linear tangent spaces. The first one, dating back to McLachlan [Mol. Phys. 8, 39–44 (1964)], minimizes the residuum of the time-dependent Schrödinger equation, while the second one, originating from the lecture notes of Kramer and Saraceno [Geometry of the Time-Dependent Variational Principle in Quantum Mechanics, Lecture Notes in Physics Vol. 140 (Springer, Berlin, 1981)], imposes the stationarity of an action functional. We characterize both principles in terms of metric and symplectic orthogonality conditions, consider their conservation properties, and derive an elementary a posteriori error estimate. As an application, we revisit the time-dependent Hartree approximation and frozen Gaussian wave packets.
The time-dependent variational principle associates to a Hamiltonian quantum system a set of trajectories running on a classical phase space with canonical equations of motion. When the quantum observables generate a Lie group G and the states are taken as functions on an appropriate homogeneous quotient space space := G/G 0 under under G , they can be equipped with a classical Poisson bracket which reproduces the commutators of the Lie group. The quantum system maps into a classical system whose equations of motion are governed by the expectation value of the quantum Hamiltonian H. We consider examples of this construction and show that the analysis of generalized classical systems provides insight into quantum many-body dynamics like chemical reactions.