A Student's Guide to the Schrödinger Equation

The objective of this work is to investigate the Schrödinger equation, analyzing the mathematical concepts employed and relating them to other areas of knowledge. In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of a physical system evolves over time. It was formulated in late 1925 and published in 1926 by the Austrian physicist Erwin Schrödinger. In quantum mechanics, the analogue of Newton's law is the Schrödinger equation for the quantum system (usually atoms, molecules, and subatomic particles are free, bound, or located). It is not a simple algebraic equation but, in general, a linear partial differential equation. The solutions to the Schrödinger equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems.

The wavenumber ν [m -1 ], represents spatial frequency which is equal to the number of wavelengths per unit distance.

Graduate Texts in Physics publishes core learning/teaching material for graduate-and advanced-level undergraduate courses on topics of current and emerging fields within physics, both pure and applied. These textbooks serve students at the MS-or PhD-level and their instructors as comprehensive sources of principles, definitions, derivations, experiments and applications (as relevant) for their mastery and teaching, respectively. International in scope and relevance, the textbooks correspond to course syllabi sufficiently to serve as required reading. Their didactic style, comprehensiveness and coverage of fundamental material also make them suitable as introductions

2019

We give an exceptionally short derivation of Schroedinger's equation by replacing the idealization of a point particle by a density distribution.

This paper attempts to summarise the basics of a radical new field in particle physics: quantum mechanics. It includes an in-depth explanation of 4 basic concepts of quantum mechanics, which are Quantum Field Theory, Deriving the Theory of Addition of Velocities or Distances, Deriving how E=mc2 was modified for massless and stationary particles and The Heisenberg Uncertainty Principle.

ResearchGate, 2021

The special problem we try to get at with these lectures is to maintain the interest of the very enthusiastic and rather smart people trying to understand physics. They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas—and spend many years studying textbooks or following online courses. Many are discouraged because there are really very few grand, new, modern ideas presented to them. Also, when they ask too many questions, they are usually told no one really understands or, worse, to just shut up and calculate. Hence, we were wondering whether or not we can make a course which would save them by maintaining their enthusiasm. This paper is the sixth chapter of such (draft) course.

183 philosophical implications of quantum mechanics and develop a new way of thinking about nature on the nanometer-length scale. This was undoubtedly one of the most signiicant shifts in the history of science. The key new concepts developed in quantum mechanics include the quantiza-tion of energy, a probabilistic description of particle motion, wave–particle duality, and indeterminacy. These ideas appear foreign to us because they are inconsistent with our experience of the macroscopic world. Nonetheless, we have accepted their validity because they provide the most comprehensive account of the behavior of matter and radiation and because the agreement between theory and the results of all experiments conducted to date has been impressively accurate. Energy quantization arises for all systems whose motions are connned by a potential well. The one-dimensional particle-in-a-box model shows why quantiza-tion only becomes apparent on the atomic scale. Because the energy level spacing is inversely proportional to the mass and to the square of the length of the box, quantum effects become too small to be observed for systems that contain more than a few hundred atoms. Wave–particle duality accounts for the probabilistic nature of quantum mechanics and for indeterminacy. Once we accept that particles can behave as waves, we can form analogies with classical electromagnetic wave theory to describe the motion of particles. For example, the probability of locating the particle at a particular location is the square of the amplitude of its wave function. Zero-point energy is a consequence of the Heisenberg indeterminacy relation; all particles bound in potential wells have nite energy even at the absolute zero of temperature. Particle-in-a-box models illustrate a number of important features of quantum mechanics. The energy-level structure depends on the nature of the potential, E n n 2 , for the particle in a one-dimensional box, so the separation between energy levels increases as n increases. The probability density distribution is different from that for the analogous classical system. The most probable location for the particle-in-a-box model in its ground state is the center of the box, rather than uniformly over the box as predicted by classical mechanics. Normalization ensures that the probability of nding the particle at some position in the box, summed over all possible positions, adds up to 1. Finally, for large values of n, the probability distribution looks much more classical, in accordance with the correspondence principle. Different kinds of energy level patterns arise from different potential energy functions, for example the hydrogen atom (See Section 5.1) and the harmonic oscil-lator (See Section 20.3). These concepts and principles are completely general; they can be applied to explain the behavior of any system of interest. In the next two chapters, we use quantum mechanics to explain atomic and molecular structure, respectively. It is important to have a rm grasp of these principles because they are the basis for our comprehensive discussion of chemical bonding in Chapter 6.

2011

odinger equation is derived classically assuming that particles present local random spatial fluctuations compatible with the presence of the zero-point field. Without specifying the forces arising from this permanent matter-field interaction but exploring its fundamental properties (homogeneity, isotropy and random aspect) to justify the emergence of the continuity equation in one-particle context, these fluctuations are described in terms of the probability density. Specifically, the starting point is the assumption that the local activities, which turn the path followed by the particle totally unpredictable, must be associated with an energy proportional to@P=@t. The polar form of the wave function, which connects the obtained classical equations with the corresponding quantum equation, emerges as a by-product of the approach.

A project for the didactic innovation and the teachers' training in quantum mechanics consists of two main phases: traditional main experiments (e.g.: Franck-Hertz, interference, polarisation); approach to Dirac to the theory. We outline a possible strategy to introduce the basic formalism of quantum mechanics without requiring an advanced mathematical or physical background. The "Dirac formulation" of Quantum Mechanics is developed by properly generalizing the description of a simple two-state system, namely the linear polarization of photons interacting with polaroids and birefringent crystals. We also discuss the relation between physical observables and linear operators, a connection which is usually considered as the "hardest" concept in Quantum Mechanics. A feasability study has been allowed by an experimentation in the fifth class of a Liceo.

Eprint Arxiv Math Ph 0505059, 2005

We expose the Schrödinger quantum mechanics with traditional applications to Hydrogen atom: the calculation of the Hydrogen atom spectrum via Schrödinger, Pauli and Dirac equations, the Heisenberg representation, the selection rules, the calculation of quantum and classical scattering of light (Thomson cross section), photoeffect (Sommerfeld cross section), quantum and classical scattering of electrons (Rutherford cross section), normal and anomalous Zeemann effect (Landé factor), polarization and dispersion (Kramers-Kronig formula), diamagnetic susceptibility (Langevin formula). We discuss carefully the experimental and theoretical background for the introduction of the Schrödinger, Pauli and Dirac equations, as well as for the Maxwell equations. We explain in detail all basic theoretical concepts: the introduction of the quantum stationary states, charge density and electric current density, quantum magnetic moment, electron spin and spin-orbital coupling in "vector model" and in the Russel-Saunders approximation, differential cross section of scattering, the Lorentz theory of polarization and magnetization, the Einstein special relativity and covariance of the Maxwell Electrodynamics. We explain all details of the calculations and mathematical tools: Lagrangian and Hamiltonian formalism for the systems with finite degree of freedom and for fields, Geometric Optics, the Hamilton-Jacobi equation and WKB approximation, Noether theory of invariants including the theorem on currents, four conservation laws (energy, momentum, angular momentum and charge), Lie algebra of angular momentum and spherical functions, scattering theory (limiting amplitude principle and limiting absorption principle), the Lienard-Wiechert formulas, Lorentz group and Lorentz formulas, Pauli theorem and relativistic covariance of the Dirac equation, etc. We give a detailed oveview of the conceptual development of the quantum mechanics, and expose main achievements of the "old quantum mechanics" in the form of exercises. One of our basic aim in writing this book, is an open and concrete discussion of the problem of a mathematical description of the following two fundamental quantum phenomena: i) Bohr's quantum transitions and ii) de Broglie's wave-particle duality. Both phenomena cannot be described by autonomous linear dynamical equations, and we give them a new mathematical treatment related with recent progress in the theory of global attractors of nonlinear hyperbolic PDEs. Namely, we suggest that i) the quantum stationary states form a global attractor of the coupled Maxwell-Schrödinger or Maxwell-Dirac equations, in the presence of an external confining potential, and ii) the wave-particle duality corresponds to the soliton-like asymptotics for the solutions of the translation-invariant coupled equations without an external potential. We emphasize, in the whole of our exposition, that the coupled equations are nonlinear, and just this nonlinearity lies behind all traditional perturbative calculations that is known as the Born approximation. We suggest that both fundamental quantum phenomena could be described by this nonlinear coupling. The suggestion is confirmed by recent results on the global attractors and soliton asymptotics for model nonlinear hyperbolic PDEs.

2009

These notes offer a basic introduction to the primary mathematical concepts of quantum physics, and their physical significance, from the operator and Hilbert space point of view, highlighting more what are essentially the abstract algebraic aspects of quantisation in contrast to more standard treatments of such issues, while also bridging towards the path integral formulation of quantisation. A discussion of the (first) Noether theorem and Lie symmetries is also included to complement the presentation. Emphasis is put throughout, as illustrative examples threading the presentation, on the quantum harmonic oscillator and the dynamics of a charged particle coupled to the electromagnetic field, with the ambition to bring the reader onto the threshold of relativistic quantum field theories with their local gauge invariances as a natural framework for describing relativistic quantum particles in interaction and carrying specific conserved charges.

1994

Centaurus, 1982

The contents of a new book of Quantum Mechanics.