Quantum mechanics with spatial non-local effects and position-dependent mass

Abstract

A new higher order Schrödinger equation characterized by a position-dependent mass is introduced based on long-range spatial kernel effects and von Roos arguments. The extended Schrödinger equation depends on the sign of the moments M k , k = 0 , 1 , 2 , … and a stabilized quantum dynamic is realized for M 2 > 0 and M 4 > 0 . We have discussed its implications in several quantum mechanical systems where more than a few were raised, mainly the emergence of the quantum Pais-Uhlenbeck and the relativistic quantum harmonic oscillators besides the complex periodic potential characterized by a PT symmetry.

A Non‐Standard Coupling Between Quantum Systems Originated From Their Kinetic Energy

“…For notable recent literature in the field, we refer to. [12][13][14][15][16][17][18][19][20][21][22][23][24] The PDM opens a new possibility, which is going to be explored in this paper. In particular, we are going to show how, using PDM, particles can be coupled through the kinetic part of the Hamiltonian.…”

Generalized uncertainty principle from long-range kernel effects: The case of the Hawking black hole temperature

“…Nevertheless, several interesting implications and outcomes have been obtained by studying the consequential 4 th -order Schrödinger equation. [15][16][17][18][19] The aims of the present study are: first, to construct a quantum diffusion equation by taking into account the longrange kernel effects which are considered in reaction-diffusion theory used in biological systems [20][21][22][23] and neutron diffusion processes [24] among others; [25][26][27] and second, to show that such a kernel approach is associated with the generalized uncertainty principle (GUP), which was introduced to account for the existence of minimal and maximal lengths in nature. [28][29][30][31][32][33][34][35][36] The presence of a minimum length suggests, in general, that space-time may be fundamentally discrete, and hence, Planck-scale corrections to the basic Heisenberg's uncertainty principle must be introduced.…”

“…To start, we introduce the following nonlocal Schrödinger integro-differential equation: [22,26,41,42]…”

“…In general, a Gaussian kernel function is used to describe quantum nonlocal diffusion processes. [26] Recall that equation ( 3) is comparable to a nonlocal diffusion equation with imaginary time. If the kernel has a second moment, then the Cauchy condition in 1D 𝑅 |x| 2 K(x)dx < +∞ holds; yet, by symmetry, the associated first moment is zero.…”