Maupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle

Abstract

Time-dependent variational principle (TDVP) provides powerful methods in solving the timedependent 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.By contrast, the TDVP gives far more practical methodologies. Indeed, there have been proposed various elaborated formalisms such as those of the so-called Dirac-Frenkel, MaLachlan [16], Kan [17], Kramer and Saraceno [18], and so on. Broeckhove, Lathouwers, Kesteloot and van Leuven [19] gave a unified account over these methods. These methods and the variants are widely used in many fields of physics and chemistry. (For more recent progress in other general theories of TDVP, we refer to [20][21][22][23][24][25][26] .) Yet, it is known that the TDVP generally faces a divergence problem in handling nonlinear variational parameters, which arises from the inversion of possibly singular matrices.An overwhelming advantage of TDVP is that it transforms the Schrödinger partial differential equation to a set of coupled ordinary differential equations over the space of variational parameters. In this context, Kan [17] determinedly states in his seminal paper that 'any time-dependent description of quantum systems derived from the variational principle is equivalent to Hamilton's description of a classical system'. On the other hand, Kan has explicitly shown that there does not exist Legendre transformation in his TDVP. Lack of the Legendre transformation suggests that there is no place for the Maupertuis reduced (or abbreviated) action [27, 28] to appear, and therefore the Maupertuis-Hamilton least action principle will not have direct significance in TDVP. It is not comfortable however to accept this assertion, since the least action principle of Maupertu...

Time-dependent variational dynamics for nonadiabatically coupled nuclear and electronic quantum wavepackets in molecules

“…With these experience, progress, and difficulty in mind, we here propose a method to quantize nuclear dynamics in terms of nuclear Gaussian wavepackets (Cartesian Gaussians multiplied by plane waves as in the squeezed states), replacing the nuclear path dynamics. The nuclear wavepackets are determined simultaneously with electron wavepackets in full quantum mechanical manner by means of the dual least action principle, a time-dependent variational principle, developed by the present author [8]. Not only the shapes and spatial distributions but also the positions of the Gaussian centers are determined quantum mechanically.…”

“…To overcome the technical matters fundamentally, and moreover, in order to explore the theoretical structure of quantum mechanics from the scope of axiomatic variational principles, we have proposed a TDVP that is based on the dual quantum mechanical Maupertuis least actions. [8] Below we apply this quantum mechanical Maupertuis-Hamilton least action principle having a set of dual variational functionals.…”

“…To address one of the ultimate goals of theoretical molecular science, we here propose to incorporate nuclear quantum wavepackets represented in the Cartesian Gaussian functions multiplied plane waves into nonadiabatic electron wavepacket dynamics we have long been developing [1][2][3][4][5][6][7] with use of our proposed quantum least action principle (time-dependent variational principle) [8]. In order to describe why the formulation of simultaneous nuclear and electronic dynamics is needed in current scientific status and to clearly distinguish the present theory from canonical and the traditional methods based on the Born-Huang expansion [10], we track our pathways to the goal starting from two early works with the Born-Huang dynamics; one is electron scattering by molecules as an electron dynamics in the fixed nuclei approximation and the a e-mail: kaztak@fukui.kyoto-u.ac.jp (corresponding author) other being direct observation of nuclear wavepacket dynamics through time-resolved photoelectron spectroscopy.…”