Symmetries in Quantum Mechanics and Statistical Physics

1999 ◽  
Vol 354 (1392) ◽  
pp. 1915-1919 ◽  
Author(s):  
Claude Allègre ◽  
Vincent Courtillot

The 20th century has been a century of scientific revolutions for many disciplines: quantum mechanics in physics, the atomic approach in chemistry, the nonlinear revolution in mathematics, the introduction of statistical physics. The major breakthroughs in these disciplines had all occurred by about 1930. In contrast, the revolutions in the so–called natural sciences, that is in the earth sciences and in biology, waited until the last half of the century. These revolutions were indeed late, but they were no less deep and drastic, and they occurred quite suddenly. Actually, one can say that not one but three revolutions occurred in the earth sciences: in plate tectonics, planetology and the environment. They occurred essentially independently from each other, but as time passed, their effects developed, amplified and started interacting. These effects continue strongly to this day.


2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Jun Tao ◽  
Peng Wang ◽  
Haitang Yang

In the framework of the deformed quantum mechanics with a minimal length, we consider the motion of a nonrelativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up toOβ2. We also show that if the slope of the potential at a turning point is too steep, the WKB connection formula is no longer valid around the turning point. The effects of the minimal length on the classical motions are investigated using the Hamilton-Jacobi method. We also use the Bohr-Sommerfeld quantization to study statistical physics in deformed spaces with the minimal length.


2014 ◽  
Vol 28 (26) ◽  
pp. 1450179 ◽  
Author(s):  
Gerhard Grössing

A research program within the scope of theories on "Emergent Quantum Mechanics" is presented, which has gained some momentum in recent years. Via the modeling of a quantum system as a non-equilibrium steady-state maintained by a permanent throughput of energy from the zero-point vacuum, the quantum is considered as an emergent system. We implement a specific "bouncer-walker" model in the context of an assumed sub-quantum statistical physics, in analogy to the results of experiments by Couder and Fort on a classical wave-particle duality. We can thus give an explanation of various quantum mechanical features and results on the basis of a "21st century classical physics", such as the appearance of Planck's constant, the Schrödinger equation, etc. An essential result is given by the proof that averaged particle trajectories' behaviors correspond to a specific type of anomalous diffusion termed "ballistic" diffusion on a sub-quantum level. It is further demonstrated both analytically and with the aid of computer simulations that our model provides explanations for various quantum effects such as double-slit or n-slit interference. We show the averaged trajectories emerging from our model to be identical to Bohmian trajectories, albeit without the need to invoke complex wavefunctions or any other quantum mechanical tool. Finally, the model provides new insights into the origins of entanglement, and, in particular, into the phenomenon of a "systemic" non-locality.


2020 ◽  
Vol 35 (11) ◽  
pp. 2050074
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi

In this paper, we use the q-derivative emerging in the non-extensive statistical physics to formulate the q-deformed quantum mechanics. We find the algebraic structure related to this deformed theory and investigate some properties of the q-deformed elementary functions. Using this mathematical background, we formulate the q-deformed Heisenberg algebra and q-deformed time-dependent Schrödinger equation. As an example, we deal with the infinite potential well and compute the Fermi energy in the q-deformed theory.


Sign in / Sign up

Export Citation Format

Share Document