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Physical Review A

Time-dependent variational principle with constraints for parametrized wave functions

Katsuhisa Ohta

Published Aug 9, 2004• 13 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.70.022503

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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.
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 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.
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.
Hamilton's principle of stationary action is formulated in classical mechanics using the Lie algebra of Schouten concomitants of symmetric contravariant tensor fields on the configuration space of the system. Such a formulation is global and coordinate free. It is shown that a directly parallel formulation holds in quantum mechanics so long as all the Poisson brackets involved can be replaced in the quantum version by commutators in a canonical way. An example (where the Hamiltonian possesses a velocity dependent potential) in which this cannot be done is discussed and concluded that in this case the action is stationary only for a subclass of variations, namely those corresponding to Killing vector fields on the configuration manifold.
To construct an NNN-representable time-dependent density-functional theory, a generalization to the time domain of the Levy-Lieb (LL) constrained-search algorithm is required. That the action is only stationary in the Dirac-Frenkel variational principle eliminates the possibility of basing the search on the action itself. Instead, we use the norm of the partial functional derivative of the action in the Hilbert space of the wave functions in place of the energy of the LL search. The electron densities entering the formalism are NNN-representable, and the resulting universal action functional has a unique stationary point in the density at that corresponding to the solution of the Schr\"odinger equation. The original Runge-Gross (RG) formulation is subsumed within the current formalism. Concerns in the literature about the meaning of the functional derivatives and the internal consistency of the RG formulation are allayed by clarifying the nature of the functional derivatives entering the formalism.