TYBSc.Phy.342.2013p.Quantum Mechanics.Sem.IV.Topic.Schrodinger equations

Olavo Leopoldino da Silva Filho, Marcello Caio Ramon e Barros Ferreira

This is the fourth paper of a series devoted to mathematically derive the Schrödinger equation using stochastic constructs. We then show that Quantum Mechanics has a stochastic support by following two paths: we first present a stochastic derivation already known in the literature and show that it is equivalent to the characteristic function derivation, made by us in the first paper of this series. However, this approach does not furnish the true stochastic dynamics of the quantum mechanical system, but only an averaged one. We then present a new derivation of the Schrödinger equation based on Langevin equations that present all features of a true stochastic system. These two approaches then improve our understanding of Quantum Mechanics to encompass stochastic behavior.

Ankit Sharma

Over the course of the past two to three decades, new tools of presentation and mathematical treatment have emerged and the subject matter of quantum mechanics has gone through significant changes. A Textbook on Modern Quantum Mechanics presents the selected elementary, intermediate, and advance topics with rejuvenated approach to the subject matter. Newly merged topics from contemporary physics and chemistry are included in the text as well as solved examples. The book covers: (i) fundamental discoveries that are the foundation of modern quantum mechanics; (ii) solution of Schrödinger’s wave equation for 1D problems and their importance; (iii) matrix and vector formulation of quantum mechanics; (iv) transformations, symmetries, and conservation laws; (v) angular and spin momenta; (vi) solution of Schrödinger equation for central potentials; (vii) time-independent perturbation theory, variational method and WKB approximation; (viii) quantum theory of scattering; (xi) many-particle systems and their quantum mechanical treatments; (x) time-dependent perturbations and the interaction of fields with matter; (xi) relativistic quantum mechanics; and (xii) quantization of fields and the second quantization. Key Features: It provides everything a student needs to know for succeeding at all levels of the undergraduate and graduate studies. It covers most of the topics that are taught under (a) elementary, (b) intermediate, and (c) advance courses of quantum mechanics at universities and colleges. It has detailed and elegant mathematical treatment with contemporary style of interpretation and presentation in simple English. Solved examples and unsolved exercises that are part of each chapter to consolidate the readers' understanding of fundamental concepts. The subject matter of the book is well tested on the students taught by the author over a period of 30 years. This is a valuable textbook for students pursuing Bachelor of Science, Master of Science, and Doctor of Philosophy (PhD) degrees in the subjects of Physics, Chemistry, and materials science in India, South Asian countries, the United States, and Europe.

Nielsen, M. P., Laugesen, Jacob, Sanjayan, J. G.

Hennadii Kovalov

Schrödinger's equation is the foundation of quantum mechanics. But it doesn't explain where phase comes from, why it can be synchronized, or what "time" is in a quantum system. The paper shows that the Schrödinger equation is a special case of chronal dynamics, where τ = ħ/E — the duration of 1 radian of phase — is considered constant. If we abandon this assumption and introduce the scalar field τ(x, t), we get a generalized equation from which we follow: Attractor E_χ = 1.41983487 MeV is the first eigenvalue of the chronal operator ("God's number") Fine-structure constant α = 1/137.036 as the ratio E_el/E_χ Electron mass m_e = (E_χ – 0.42)/2 Hierarchy of leptons through scaling χ = φ·τ Synchronization condition cos φ = 1, K = 0.316, Δω/ω = 0.2 Keywords: Schrödinger equation, chronal dynamics, τ = ħ/E, χ = φ·τ, attractor, God number.