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The G-dynamics of the QCPB theory

Gen Wang

Published Mar 31, 2020• 1 citation
OpenAlex

Abstract

In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the QCPB theory. Then we want to seek the precise formulations of the G-dynamics to calculate the practical quantum problems. We explicitly verify how the role of G-dynamics plays in quantum mechanics, and then we propose three new operators based on the G-dynamics: D-operator, T-operator and G-operator, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then the eigenvalue equation of them are discussed accordingly by using the Schrodinger equation and geometrinetic energy operator (GEO) which leads to the Ri-operator, respectively, we obtain some useful results between two quantum situations. Meanwhile, we present generalized covariant wave equation based on the G-operator. The GEO for quantum harmonic oscillator as an application is further calculated.

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New U -matrix theory in quantum mechanics

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quant-pharXiv:2606.02112•Jun 2, 2026•0 citations
In this paper, we further study the G-dynamics newly emerged in the covariant dynamics defined by the QCPB theory. Then we want to seek the precise formulations of the G-dynamics to calculate the practical quantum problems. We explicitly verify how the role of G-dynamics plays in quantum mechanics, and then we propose three new operators based on the G-dynamics: D-operator, T-operator and G-operator, we find that the non-Hermitian operators are inevitable to appear in the framework of the QCPB. Then the eigenvalue equation of them are discussed accordingly by using the Schrodinger equation and geometrinetic energy operator (GEO) which leads to the Ri-operator, respectively, we obtain some useful results between two quantum situations. Meanwhile, we present generalized covariant wave equation based on the G-operator. The GEO for quantum harmonic oscillator as an application is further calculated.
Using the quantum covariant Poisson bracket (QCPB) theory, we can accomplish much more compatible explanations of the quantum mechanics supported by the G-dynamics. We further study the generalized quantum harmonic oscillator equipped with the G-dynamics of type I, such as geometric creation and annihilation operators, and the geometric number operator as an extension of the number operator is well given for the deep discussions, the geometric Hamiltonian operator is expressed as another form. Especially, the geometric quantization rules based on the QCPB theory is then calculated.
Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a generalized Hamiltonian dynamics and to a covariant Schrödinger type equation is shown. Examples for relativistic potential problems are solved. Mathematically and physically acceptable probability densities for the Klein-Gordon equation and for the Dirac equation are derived, and a new continuity equation for each case is given. The quantum distribution for mass is discussed, and unambiguous representations of four-velocity and four-acceleration operators are given.
Quantum covariant Hamiltonian system theory provides a coherent framework for modelling the complex dynamics of quantum systems. In this paper, we centrally deal with the Hermiticity of quantum operators that directly links to the physical observable, thusly, we give a rigorous proof to verify one-dimensional G-dynamics w^(cl)=w^(cl)†∈Her{\hat{w}^{\left( cl \right)}}={\hat{w}^{\left( cl \right)\dagger }}\in Herw^(cl)=w^(cl)†∈Her that is a Hermitian operator satisfying (w^(cl)ϕ,φ)=(ϕ,w^(cl)φ)\left( {\hat{w}^{\left( cl \right)}}ϕ,φ\right)=\left( ϕ,{\hat{w}^{\left( cl \right)}}φ\right)(w^(cl)ϕ,φ)=(ϕ,w^(cl)φ) for any two states ϕϕϕ and φφφ, and its eigenvalues are real. We also prove that curvature operator is a skew-Hermitian operator as well. The act of finishing this Hermitian proof valuably enables us to ensure the non-Hermitian Hamiltonian operator H^(ri)=H^(g)−H^(clm⁡)∈NHer{\hat{H}^{\left( ri \right)}} ={\hat{H}^{\left( g \right)}} -{\hat{H}^{\left( \operatorname{clm} \right)}}\in NHerH^(ri)=H^(g)−H^(clm)∈NHer that is divided into the Hermitian operator H^(g)=H^(cl)−E(s)/2∈Her{\hat{H}^{\left( g \right)}} ={\hat{H}^{\left( cl \right)}}-{{E}^{\left( s \right)}}/2\in HerH^(g)=H^(cl)−E(s)/2∈Her and the skew-Hermitian operator H^(clm⁡)=−1ℏw^(cl)∈SHer{\hat{H}^{\left( \operatorname{clm} \right)}}=\sqrt{-1}\hbar {\hat{w}^{\left( cl \right)}}\in SHerH^(clm)=−1​ℏw^(cl)∈SHer generally, and H^(ri){\hat{H}^{\left( ri \right)}}H^(ri) always has the complex eigenvalues. We use the formula of the G-dynamics to evaluate the Berry-Keating's Hamiltonian operator H^(bk)=−−1ℏθ^/2∈Her{\hat{H}^{\left( \text{bk}\right)}}=-\sqrt{-1}\hbar \hat{θ}/2\in HerH^(bk)=−−1​ℏθ^/2∈Her and its extensive version H^(gbk)∈NHer\hat{H}^{\left( \text{gbk}\right)}\in NHerH^(gbk)∈NHer as the applications of the G-dynamics, to see how the similarity appears in the light of obvious factor θ^/2=xddx+1/2∈SHer\hat{θ}/2=x\frac{d}{dx}+1/2\in SHerθ^/2=xdxd​+1/2∈SHer, etc.
We have analyzed Dyson's UUU-matrix theory of solving the Schr\"odinger equation in the interaction picture and are able to express the UUU matrix as a dominant term plus an infinite series involving multiple integrals of time. For a certain rather restrictive class of Hamiltonians, our theory is exact for a general time-dependent problem. For other Hamiltonians, we can only obtain approximate expressions for our UUU matrix and hence the wave function. Treating a time-independent problem as a special case of the time-dependent situation with a sudden-switching process, we have shown that our UUU matrix is exact. To demonstrate the working procedures of our theory, we apply it to study the well-known time-independent charged harmonic-oscillator problem and the more general harmonic oscillator with a time-dependent driving force. Compared with other methods, our new theory appears to lead to a result which contains more information than others due to the inclusion of noncommutability properties of operators in the operator Schr\"odinger equation. It has been shown that the classical Feynman path-integral formalism can be deduced from quantum mechanics with the use of the Green's-function operator. It is interesting to note that apart from a step function, the Green's-function operator is the same as that of our U(s){U}^{(s)}U(s) matrix, which is the UUU matrix obtained within the regime of the Schr\"odinger picture for a time-independent Hamiltonian, as a special case of our general time-dependent treatment.
We investigate the quantum dynamics of a particle subjected to a linear potential using the Lewis--Riesenfeld invariant operator method. Starting from the time-dependent Schrödinger equation associated with a constant external force, we construct the most general Hermitian quadratic invariant and derive the corresponding coupled differential equations for its time-dependent coefficients. By means of an appropriate sequence of unitary transformations, the invariant operator is reduced to the form of a harmonic oscillator Hamiltonian. This reduction enables a clear classification of the system according to the sign of the conserved quantity ω2. Particular attention is devoted to the physically relevant case ω2 >0, which yields a discrete eigenspectrum. Explicit analytical expressions for the invariant coefficients, the displacement parameters, and the transformed wave functions are obtained. The resulting formalism provides an exact quantum description of a particle under a constant force and establishes a direct connection between invariant theory and harmonic oscillator quantization.