Foundations of Quantum Mechanics in Hilbert and Phase Space with Application to Nuclear Fission.

Wawrzyniec Guz

Shimon Levit, John W. Negele, Zvi Paltiel

A mean-field theory is obtained for the spontaneous decay of unstable nuclei by applying the stationary-phase approximation to a functional integral expression for $\mathrm{Tr}{(H\ensuremath{-}E)}^{\ensuremath{-}1}$. The method is applied first to the lifetime of a metastable state in one-dimensional quantum mechanics and subsequently generalized to many-fermion systems. Solutions to the resulting nonlinear self-consistent field equations are presented for a self-bound, saturating many-body system of fermions in one spatial dimension.NUCLEAR REACTIONS, FISSION Derivation of dynamical mean-field equations for spontaneous fission, using imaginary time stationary-phase approximation to a functional integral expression for $\mathrm{Tr}{(H\ensuremath{-}E)}^{\ensuremath{-}1}$. Application to a many-fermion model in one spatial dimension.

Syed Muntasir Mamun

Hilbert spaces, introduced in the early twentieth century as infinite-dimensional generalisations of Euclidean space, provide the rigorous mathematical framework for quantum mechanics and numerous applied disciplines. This paper offers a comprehensive examination of their historical development, axiomatic structure, and multifaceted applications, with particular emphasis on their foundational role in quantum mechanics. We then connect these structures to quantum information geometry, where the (projective) Hilbert space is endowed with natural Riemannian metrics—most notably the Fubini–Study metric for pure states and the Bures metric for mixed states—that quantify statistical distinguishability, parameter estimation precision, and geometric phases. By exploring classical information geometry as a precursor and highlighting quantum extensions, we illustrate how Hilbert-space geometry reveals deep insights into quantum phase transitions, metrology, and quantum technologies. Nuances such as separability, rigged Hilbert spaces for continuous spectra, and infinite-dimensional subtleties are addressed, alongside implications for machine learning via reproducing kernel Hilbert spaces (RKHS) and beyond. The analysis underscores the enduring interplay between pure mathematics and quantum physics. Mamun, SM (2026) …………………………………..1 Hilbert Spaces: Foundations, Applications, and Geometric Connections to Quantum Information Theory (Vol-I) Also available on Academia.edu: https://www.academia.edu/attachments/132277960/download_file?s=portfolio

David A. Edwards

Paul Carpentier, N. Pillet, Denis Lacroix A +2 more

Several protocols are proposed to build continuous energy surfaces of many-body quantum systems, regarding both energy and states. The standard variational principle is augmented with constraints on state overlap, ensuring arbitrary precision on continuity. As an illustration, the lowest energy and excited state paths relevant for the ^{240}Pu asymmetric fission are studied. The scission is clearly signed, with a neutron excess in the neck, the ultimate glue before its rupture. Our approach can potentially connect any couple of Hilbert space states, which opens up new horizons for various applications.

Neumann, J. v., Wheeler, N. A.