It's a very important content in quantum mechanics to solve the problems of potential wells or potential hills. The deep acknowledgement and accurate application of the boundary conditions are an important link to the solution of the problems of this kind. But most of the current quantum mechanics textbooks have not provided a complete reasoning to the boundary conditions. They only directly provide the corresponding boundary condition in the solution of a problem or make an explanation in words according to its physics meaning Aiming at the above insufficiency, this article deduces the boundary conditions from the point of mathematics and simply discusses its scope of application.
It's a very important content in quantum mechanics to solve the problems of potential wells or potential hills. The deep acknowledgement and accurate application of the boundary conditions are an important link to the solution of the problems of this kind. But most of the current quantum mechanics textbooks have not provided a complete reasoning to the boundary conditions. They only directly provide the corresponding boundary condition in the solution of a problem or make an explanation in words according to its physics meaning Aiming at the above insufficiency, this article deduces the boundary conditions from the point of mathematics and simply discusses its scope of application.
Quantum mechanics is one of the most challenging subjects to learn. It is challenging because quantum phenomenon is counterintuitive, and the mathematics used to explain such a phenomenon is very abstract, and difficult to grasp. This textbook is an attempt to overcome these challenges. Every chapter presents quantum ideas step- by- step in a structured way with a comparison between quantum and classical concepts. It provides a clear distinction between classical and quantum logic. Conceptual questions are provided after every important section so that the reader can test their understanding at every step. Such an approach aids in preventing misconceptions. Problem solving is not restricted to solving differential equations and integration. But it requires to systematically and creatively analyze a problem, to apply the new and powerful concepts for finding a solution and to understand the physical meaning of the solution. The tutorials on special topics are an effort to teach problem solving by actively engaging the reader in a thinking process, to apply the concepts and to understand the physical meaning of the solution. The simulations are provided for some of the topics. The simulations aid in the visualization of the quantum phenomenon, and for meaningful understanding of the mathematics. This approach may lead to development of "quantum mechanical intuition "as well as learning mathematical techniques for problem solving. Most importantly, the book is not flooded with numerous topics that makes the reader confused and distracted, rather the most important topics are discussed at a deeper level. The understanding of quantum mechanics is incomplete without understanding the early ideas and experiments that lead to the development of the quantum theory. Thus, the first two chapters of the book are dedicated to such topics. The key features of this book are: A simplified, structured, and step-by-step introduction to quantum mechanics. The simplification is attained through use of two-level system, step- by- step discussion of important topics in a simplified language at a deeper level, analogies, and visualization using illustrations and simulations A systematic arrangement of topics, and numerous worked- out examples. The presentation of the structure in the mathematical formalism of quantum mechanics provides clarity in understanding complicated and abstract mathematics. It also helps to understand the distinction between the quantum mechanical and classical approaches Conceptual questions at the end of every important section. The conceptual questions can be used in a classroom as a point of discussion between an instructor and students Tutorials on special topics. Simulations on special topics aid in the visualization of the physical phenomenon, and demonstration of the application of mathematics An in-depth discussion of the wave-particle duality, measurement problem, and their philosophical implications in Chapter 2 provides an understanding of the broader meaning of quantum mechanics
Reasons are given for believing that the boundary conditions at a discontinuity in the potential (V) in wave mechanics may be ${\ensuremath{\varphi}}_{1}={\ensuremath{\varphi}}_{2}{\ensuremath{\mu}}^{\frac{1}{2}}$ and $\frac{{\ensuremath{\varphi}}_{1}d{\ensuremath{\varphi}}_{1}}{\mathrm{dx}}=\frac{{\ensuremath{\varphi}}_{2}d{\ensuremath{\varphi}}_{2}}{\mathrm{dx}}$. Here $\ensuremath{\varphi}$ is the wave function and $\ensuremath{\mu}$ the ratio of the group velocities. With these boundary conditions the waves are either completely transmitted or completely reflected according as the energy E is greater or less than V. The particles therefore need not be supposed to enter the region in which V>E where their velocities would be imaginary and their kinetic energies negative. It is suggested that the group of waves associated with a particle should be identified with the particle and that the energy and momentum of the particle are the energy and momentum of the group of waves. When the group is stopped or absorbed it contracts to small dimensions and so becomes a particle.
A key element of theoretical physics is the conceptualisation of physical phenomena in terms of models, which are then investigated by the tools at hand. For quantum many-body systems, some models can be exactly solved, i.e., their physical properties can be calculated in an exact fashion. There is often a deep underlying reason why this can be done—the theory of integrability—which manifests itself in many guises. In Beautiful models , Bill Sutherland looks at exactly solved models in quantum many-body systems, a well established field dating back to Bethe's 1931 exact solution of the spin-1/2 Heisenberg chain. This field is enjoying a renaissance due to the ongoing and striking experimental advances in low-dimensional quantum physics, which includes the manufacture of quasi one-dimensional quantum gases. Apart from the intrinsic beauty of the subject material, Beautiful Models is written by a pioneering master of the field. Sutherland has aimed to provide a broad textbook style introduction to the subject for graduate students and interested non-experts. An important point here is the `language' of the book. In Sutherland's words, the subject of exactly solved models `belongs to the realm of mathematical physics—too mathematical to be `respectable' physics, yet not rigorous enough to be `real' mathematics. ...there are perennial attempts to translate this body of work into either respectable physics or real mathematics; this is not that sort of book.' Rather, Sutherland discusses the models and their solutions in terms of their `intrinisic' language, which is largely as found in the physics literature. The book begins with a helpful overview of the contents and then moves on to the foundation material, which is the chapter on integrability and non-diffraction. As is shown, these two concepts go hand in hand. The topics covered in later chapters include models with δ-function potentials, the Heisenberg spin chain, the Hubbard model, exchange models, the Calogero–Sutherland models and models with ground state wavefunctions of product form. One of my favourites is chapter 7, dealing with the consistency conditions for two-body scattering operators involving the Yang-Baxter equations. The strength of the book lies in the pedagogical approach, with the underlying emphasis on integrability and the Bethe ansatz. Sutherland brings his own insights to these problems, and as such there is also something to be gained by specialist readers. Given the author's aims, the book is not meant to be a complete and historic review of the field. Rather, the intention is that the general references given will point the reader in the right direction. Sutherland quite wisely gives an up-front apology to any authors he has left out. Certainly such historical omissions can be found. Two obvious examples are the origins of the condition of non-diffraction and twisted boundary conditions. Setting such minor points aside, Beautiful Models is a most welcome book. It does a great service to a fascinating, enduring and increasingly relevant field by highlighting not only the beauty, but also the magic of integrability.
The one-dimensional finite-depth potential well problem under certain boundary conditions discussed in this paper is mainly the eigenvalue problem in quantum mechanics. The basis of this article is the solution of Schrodinger equation and the treatment of transcendental equation. In many quantum mechanics textbooks, the one-dimensional infinite potential well problem is regarded as the foundation of quantum mechanics because it is easy to handle and understand. On this basis, this paper discusses the more challenging finite-depth potential well situation. In this paper, the transcendental equation is obtained by analogy between the wave function with time and the continuity of the boundary conditions. In order to solve the problem that transcendental equation is difficult to solve in this process, this paper adopts the method of solving numerical solutions by using images, which includes two different cases of odd and even functions. This paper reveals the possibility of particles appearing outside the well in this problem, and on this basis, some discussions and analysis are made on the one-dimensional asymmetric finite-depth potential well problem.