It is impossible for students to master the basic ideas and conclusions of quantum mechanics without solving a certain set of problems. Problems play a significant role in the educational process, forming not only logical thinking, but also influencing the overall development of the students. The article poses and solves problems about stationary one-dimensional problems in quantum mechanics.
It is impossible for students to master the basic ideas and conclusions of quantum mechanics without solving a certain set of problems. Problems play a significant role in the educational process, forming not only logical thinking, but also influencing the overall development of the students. The article poses and solves problems about stationary one-dimensional problems in quantum mechanics.
This collection of solved problems corresponds to the standard topics covered in established undergraduate and graduate courses in Quantum Mechanics. Problems are also included on topics of interest which are often absent in the existing literature. Solutions are presented in considerable detail, to enable students to follow each step. The emphasis is on stressing the principles and methods used, allowing students to master new ways of thinking and problem-solving techniques. The problems themselves are longer than those usually encountered in textbooks and consist of a number of questions based around a central theme, highlighting properties and concepts of interest. For undergraduate and graduate students, as well as those involved in teaching Quantum Mechanics, the book can be used as a supplementary text or as an independent self-study tool.
This book contains more than 300 problems in quantum mechanics with accompanying solutions, covering topics that are commonly taught in first-year graduate physics programs. Special care is given to each problem's formulation, with detailed and extensive solutions provided to support understanding. The problems span a range of difficulties, from basic exercises to more challenging applications and extensions of the standard material. Students are required to think critically and incorporate physics and mathematical techniques learned previously or concurrently to solve the more challenging problems. Each chapter begins by framing the particular topic being examined with a short theory section that sets the context for and motivates the problems that follow. This text is well suited for self-study or as a useful supplement to the existing quantum mechanics textbooks for upper-undergraduate and graduate students, and their instructors.
The goal of Quantum Mechanics I, A Problem Text, is to enable students to solve problems appropriate to their first course in quantum mechanics. Significantly detailed solutions to traditionally posed problems dominate. Theory and rationale are addressed in prescript and postscript narratives and/or within the solved problems to guide students; and clarify vocabulary, symbology, and terminology. Applications of the postulates are presented initially using arguments from linear algebra appropriate to discrete systems with low-dimensional vectors and matrix operators, introducing orthogonality, orthonormality, Hermiticity, eigenvectors, eigenvalues, probability, expectation value, and degeneracy. These arguments blend into infinite-dimensional vectors and operators, and calculus-based arguments for continuous systems where delta functions, theta functions, and Fourier transforms are initially discussed. Dirac notation is fully developed within these first five chapters. Chapter six discusses Ehrenfest’s theorem, the Heisenberg uncertainty relations, and Gaussian wave functions and wave packets. Increasingly sophisticated potential energy functions are then treated to include the free particle, the infinite square well, the one-dimensional scattering state, the simple harmonic oscillator, the finite square well, orbital angular momentum, and the hydrogen atom. Ladder operators are featured in the developments of the simple harmonic oscillator and orbital angular momentum. The diversity of narratives and detailed solutions intend
Formation of quantum mechanics bases is impossible without acquisition of knowledge and abilities for the solution of the corresponding tasks in this section of theoretical physics. At the same time, the exact solution of tasks is not always possible in quantum mechanics, especially if they are non-stationary and multipartial tasks. Various approximate methods (the theory of perturbations, a variation method, quasiclassical and adiabatic approximations) are developed for the solution of quantum mechanical problems. Sometimes tasks which are studied in a standard course of quantum mechanics can be solved by the one-dimensional equation of Schrodinger, but the division method of variables is, in fact, exclusive, and similar decisions can have special features, which are not characteristic for solutions of a general kind. The quantity of the solution methods applied to this class of problems is limited. In this article it is shown that a number of the main representations of quantum mechanics can be created without preliminary studying of its mathematical apparatus, by use of de Broil waves.
This paper presents a way to use a computer in the teaching of introductory, stationary state, Schrödinger quantum mechanics. The method is based upon an easily explained computer method; a 20-line program in BASIC is presented to implement the strategy. Not only is this numerical technique completely general for all one-dimensional and all spherically symmetric, three-dimensional problems, but also it allows the student to be introduced to topics of interest such as asymptotic solutions. Several examples of the use of the computer technique are presented.
A method is proposed for the quantum-mechanical determination of the eigenstates and eigenvalues of a particle (e.g. an electron) in a potential of general shape. For such systems the method allows one to calculate the exact solutions of the stationary Schrödinger equation. For the application of this method any well-known reference system is needed. A transition from the stationary reference system to the stationary system under investigation is performed by means of the time-dependent Schrödinger equation. The results are compared with various standard methods; alternative numerical methods will be discussed critically.