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

Abstract

In this work we consider two conditions required for the nonsingularity of constraints in the timedependent 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. Especially for analytic wave functions for complex TDVPparameters, the regularity and the second-class conditions become equivalent. The second-class condition for expectation values is reduced to the non-commutability of the corresponding quantum operators. The symplectic singularity of the equation of motion of TDVP (TDVP-EOM) is also shown to be a local breakdown of the second-class condition in an extended canonical phase-space.

Constraints in the variational principle for stationary states

“…By Eq. ( 14), the VPSS constraints operators { Ĝi } i=1,M are classified as first-class TDVP operators [6][7][8][9] in the form of the Hypervirial Theorem [22,23]. The VPSS has already been derived so that trajectories do not evolve with time within the framework of the TDVP [7,8].…”

“…Constraints in the variational principle for stationary states (VPSS) [1] can be used for various purposes, such as to keep some formal symmetries of the system, to construct models of some physical situations, and to analyze physical or chemical consequences of some freedoms [2][3][4][5]. In previous works [6][7][8][9], by using the pseudo-classical structure of the timedependent variational principle (TDVP) [10][11][12][13][14][15][16][17], we have systematically analyzed nonsingular constraints in the TDVP in accordance with Dirac's constrained classical mechanics [18][19][20][21].…”