Effective equations in mathematical quantum mechanics

Boris Khots, Dmitriy Khots

Certain results that have been predicted by Quantum Mechanics (QM) theory are not always supported by experiments. This defines a deep crisis in contemporary physics and, in particular, quantum mechanics. We believe that, in fact, the mathematical apparatus employed within today's physics is a possible reason. In particular, we consider the concept of infinity that exists in today's mathematics as the root cause of this problem. We have created Observer's Mathematics that offers an alternative to contemporary mathematics. This paper is an attempt to relay how Observer's Mathematics may explain some of the contradictions in QM theory results. We consider the Hamiltonian Mechanics, Newton equation, Schrodinger equation, two slit interference, wave-particle duality for single photons, uncertainty principle, Dirac equations for free electron in a setting of arithmetic, algebra, and topology provided by Observer's Mathematics (see www.mathrelativity.com). Certain results and communications pertaining to solution of these problems are provided.

Dawit Hiluf Hailu

In this paper we introduce the use of Sylvester's formula for systems with degenerate eigenvalues in relation to obtaining their analytical solutions. To appreciate the use we include two other forms of analytical solutions namely adiabatic and Magnus approximations. In quantum mechanics, the Schrödinger equation is a mathematical equation that describes the evolution over time of a physical system in which quantum effects, such as wave--particle duality, are significant. The equation is a mathematical formulation for studying quantum mechanical systems. Just like Newtons's laws govern the motion of objects, Schrödinger equations of motion also govern the motion of quantum objects. Unlike the classical motion of objects the equation of motions of quantum phenomenon deals with the likelihood of the trajectories.

S. G. Eckstein, B. B. Varga, S. Ø. Aks

Walter E Thirring

Pedro L. Garrido, Sheldon Goldstein, Jani Lukkarinen +1 more

We discuss a phenomenon of elementary quantum mechanics that is counterintuitive, non-classical, and apparently not widely known: the reflection of a particle at a downward potential step. In contrast, classically, particles are reflected only at upward steps. The conditions for this effect are that the wavelength is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. The phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation. We present numerical and mathematical evidence that it is also predicted by the time-dependent Schrödinger equation. The paradoxical reflection effect suggests and we confirm mathematically that a particle can be trapped for a long time (though not indefinitely) in a region surrounded by downward potential steps, that is, on a plateau.