Open AccessMaupertuis-Hamilton least action principle in the space of variational parameters for Schrödinger dynamics; A dual time-dependent variational principle
Author(s) -
Kazuo Takatsuka
Publication year - 2020
Publication title -
journal of physics communications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.407
H-Index - 17
ISSN - 2399-6528
DOI - 10.1088/2399-6528/ab7b34
Subject(s) - principle of least action , variational principle , legendre transformation , action (physics) , schrödinger's cat , hamilton's principle , mathematics , quantum , classical mechanics , calculus of variations , mathematical physics , physics , quantum mechanics , mathematical analysis , equations of motion
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.