Related papers: The quantum Arnold transformation
We discuss an alternative version of non- relativistic Newtonian mechanics in terms of a real Hilbert space mathematical framework. It is demonstrated that the physics of this scheme correspondent with the standard formulation.…
The formulation of classical mechanics applicable to fermionic degrees of freedom is presented in mathematically rigorous terms, including a description of how the mathematical structure relates to the quantization of the theory. Canonical…
Considered is the Schr\"odinger equation in a finite-dimensional space as an equation of mathematical physics derivable from the variational principle and treatable in terms of the Lagrange-Hamilton formalism. It provides an interesting…
We expose the Schr\"odinger quantum mechanics with traditional applications to Hydrogen atom. We discuss carefully the experimental and theoretical background for the introduction of the Schr\"odinger, Pauli and Dirac equations, as well as…
We show that, in spite of a rather common opinion, quantum mechanics can be represented as an approximation of classical statistical mechanics. The approximation under consideration is based on the ordinary Taylor expansion of physical…
A formulation of quantum mechanics is introduced based on a $2D$-dimensional phase-space wave function $\text{\reflectbox{\text{p}}}\mkern-3mu\text{p}\left(q,p\right)$ which might be computed from the position-space wave function…
We use tools from non-standard analysis to formulate the building blocks of quantum field theory within the framework of categorical quantum mechanics. Building upon previous work, we construct an object of *Hilb having quantum fields as…
The quantum dynamic equation (QDE) of machine learning is obtained based on Schr\"odinger equation and potential energy equivalence relationship. Through Wick rotation, the relationship between quantum dynamics and thermodynamics is also…
Based on a number of experimentally verified physical observations, it is argued that the standard principles of quantum mechanics should be applied to the Universe as a whole. Thus, a paradigm is proposed in which the entire Universe is…
Geometric (Schrodinger) quantization of nonrelativistic mechanics with respect to different reference frames is considered. In classical nonrelativistic mechanics, a reference frame is represented by a connection on a configuration space…
We consider quantum systems which interact strongly with a rapidly varying environment and derive a Schrodinger-like equation which describes the time evolution of the average wave function. We show that the corresponding Hamiltonian can be…
We consider the relations between nonstationary quantum oscillators and their stationary counterpart in view of their applicability to study particles in electromagnetic traps. We develop a consistent model of quantum oscillators with…
Starting from generalized position operators, we derive complex and quaternionic angular momentum operators along with their commutation algebra as well. These algebras differ from the standard Hermitian ones, especially in terms of…
This paper presents the geometric setting of quantum variational principles and extends it to comprise the interaction between classical and quantum degrees of freedom. Euler-Poincar\'e reduction theory is applied to the Schr\"odinger,…
Using a new state-dependent, $\lambda$-deformable, linear functional operator, ${\cal Q}_{\psi}^{\lambda}$, which presents a natural $C^{\infty}$ deformation of quantization, we obtain a uniquely selected non--linear, integro--differential…
In static classical statistical systems the problem of information transport from a boundary to the bulk finds a simple description in terms of wave functions or density matrices. While the transfer matrix formalism is a type of Heisenberg…
We consider the quantum version of Arnold's generalisation of a rigid body in classical mechanics. Thus, we quantise the motion on an arbitrary Lie group manifold of a particle whose classical trajectories correspond to the geodesics of any…
Semiclassical approximations to quantum dynamics are almost as old as quantum mechanics itself. In the approach pioneered by Wigner, the evolution of his quasiprobability density function on phase space is expressed as an asymptotic series…
The formalism of classical and quantum mechanics on phase space leads to symplectic and Heisenberg group representations, respectively. The Wigner functions give a representation of the quantum system using classical variables. The…
It is well-known that time-dependent Schr\"{o}dinger equation can only be exactly solvable in very rare cases, even for two-level quantum systems. Therefore, finding exact quantum dynamics under time-dependent Hamiltonian is not only of…