Related papers: Quantum mechanics and umbral calculus
By assuming that the kinetic energy,potential energy,momentum,and some other physical quantities of a particle exist in the field out of the particle,the Schrodinger equation is an equation describing field of a particle,but not the…
We obtain exact solutions to the class of parabolic partial differential equations of arbitrary dimensionality and with arbitrary potentials. The solutions are presented in a compact-form: as explicit mathematical expressions consisting of…
At present, our notion of space is a classical concept. Taking the point of view that quantum theory is more fundamental than classical physics, and that space should be given a purely quantum definition, we revisit the notion of Euclidean…
Classical mechanics involves position and momentum variables that must be special coordinates chosen to promote to suitable quantum operators. Since classical variables may be broadly chosen, only unique variables should be chosen. We will…
The linear Schr\"odinger equation with piecewise constant potential in one spatial dimension is a well-studied textbook problem. It is one of only a few solvable models in quantum mechanics and shares many qualitative features with…
We apply the principles discussed in an earlier paper to the construction of discrete time field theories. We derive the discrete time field equations of motion and Noether's theorem and apply them to the Schrodinger equation to illustrate…
In classical physics, there is a basic principle, namely "A particle cannot be located at the position of another one on the same time". Which consequences can be derived if this principle is transferred into quantum physics? For doing…
Owing to their long-lifetimes at cryogenic temperatures, mechanical oscillators are recognized as an attractive resource for quantum information science and as a testbed for fundamental physics. Key to these applications is the ability to…
The quantum search algorithm is a technique for searching N possibilities in only sqrt(N) steps. Although the algorithm itself is widely known, not so well known is the series of steps that first led to it, these are quite different from…
In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many of the properties considered for shift invariant difference operators satisfying the…
The numerical integration of the Schr\"odinger equation by discretization of time is explored for the curved manifolds arising from finite representations based on evolving basis states. In particular, the unitarity of the evolution is…
The construction of a consistent theory of quantum gravity is a problem in theoretical physics that has so far defied all attempts at resolution. One ansatz to try to obtain a non-trivial quantum theory proceeds via a discretization of…
We study the quantum mechanics of the derivative nonlinear Schrodinger equation which has appeared in many areas of physics and is known to be classically integrable. We find that the N-body quantum problem is exactly solvable with both…
Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like…
The Schr\"odingerisation method combined with the autonomozation technique in \cite{cjL23} converts general non-autonomous linear differential equations with non-unitary dynamics into systems of autonomous Schr\"odinger-type equations, via…
A method for obtaining discretization formulas for the derivatives of a function is presented, which relies on a generalization of divided differences. These modified divided differences essentially correspond to a change of the dependent…
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shr\"odinger propagator…
Since a pure quantum system is incapable of faithfully simulating the solutions of the Schroedinger equation that actually pertains to itself, it is proposed that quantum computing technology (as opposed to cryptographic technology) not be…
We propose six principles as the fundamental principles of quantum mechanics: principle of space and time, Galilean principle of relativity, Hamilton's principle, wave principle, probability principle, and principle of indestructibility and…
In relativistic quantum field theory with local interactions, charge is locally conserved. This implies local conservation of probability for the Dirac and Klein-Gordon wavefunctions, as special cases; and then in turn for non-relativistic…