FORMATION OF THE MAIN REPRESENTATIONS ABOUT QUANTUM MECHANICS
Igor Viktorovich Ryzhov, Рыжов Игорь Викторович, Косова Ирина Святославовна, Косова Ирина Святославна, Nikolay Aleksandrovich Vasiliev, Васильев Николай Александрович, L. V. Zhukov, Жуков Лев Викторович, Vladimir Nikolayevich Aniskin, Vladimir Nikolaevich Aniskin
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.
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.
The book contains 800 problems mainly on non-relativistic quantum mechanics, covering both the basic provisions and mathematical apparatus of the theory, as well as its numerous applications to atomic physics, nuclear and particle physics and physics of multiparticle systems. Considerable attention is paid to the computational methods of quantum mechanics: perturbation theory, quasiclassical approximation, variational method, sudden impact approximation, adiabatic method. Two chapters (out of a total of 16) include problems on elementary quantum radiation theory and relativistic wave equations. These problems are a useful introduction to the study of quantum field theory and particle physics.
The solution of most problems associated with electron quantum states in physical systems and structures (atoms, molecules, quantum nanostructure objects, and crystals) is hard to find because of the mathematical difficulties of getting exact solutions of the Schrödinger equation. Therefore, approximate methods of solving such problems are of special interest. We will consider some of these methods, such as the adiabatic approximation now and later the effective-mass method , using real physical systems as examples. In this chapter we will consider several widely used approximation methods for finding the wavefunctions and energies of quantum states as well as the probabilities of transitions between quantum states. First of all, we will consider stationary and non-stationary perturbation theories. What is common to these two theories is that it is assumed that the perturbation is weak and that it changes negligibly the state of the unperturbed system. Stationary perturbation theory is used for the approximate description of a system's behavior if the Hamiltonian of the quantum system being considered does not directly depend on time. In the opposite case, non-stationary theory is used. Then, we will briefly consider the quasiclassical approximation, which is used for the problems of quantum mechanics which are close to analogous problems of classical mechanics. Stationary perturbation theory for a system with non-degenerate states This theory is used for the approximate calculation of the energy levels and the wavefunctions of stationary states of systems that are subjected to the influence of small perturbations.
In quantum mechanics, there exist a lot of problems from the domain of atomic, molecular, solid-state physics, physics of magnetic phenomena, nuclear physics, etc. The number of problems which may be solved exactly are few such as some one-dimensional problems, problems in the rectangular well (barrier), the oscillator, problems of motion in a central field such as motion in a Coulomb field. These methods are well formulated and are used in evaluations to a greater approximation. However, in some cases, these may create problems that are very difficult, if not impossible, to be solved exactly. In such problems, the Hamiltonian cannot be exactly diagonalized in a simple manner. For such problems, we resort to approximate methods for finding the eigenvalues and eigenfunctions of the Hamiltonian. Besides, these approximate methods yield results closer to experimental ones. Amongst the approximate methods, we examine the variational method.
Abstract This book provides the reader with a contemporary and comprehensive introduction to Quantum Mechanics. It is suitable for beginners as well as for more advanced university students. Quantum mechanics is presented in a pedagogical fashion, with a clear logical organization. The various concepts and methods are introduced first in elementary terms, and later developed into more precise formulations. Systematic studies of approximation methods and the discussion of a wide class of physical applications follow. Part I of the book, together with the opening sections of Part II, provide adequate material for an introductory course of one semester at most universities. The rest of the book might be used in an advanced course on Quantum Mechanics. The basic material is fairly standard, even though some discussions such as those on general systems with time-dependent Hamiltonians, on metastable systems, as well as the discussions in some of the Complement sections, may not be found in other textbooks. The book also contains many original observations or new ways of illustrating even well-known subjects. In fact, the authors wish to convey in this book the sense of wonder in the logical simplicity and at the same time the beauty of subtle and far-reaching consequences of Quantum Mechanics, to young physics students in particular. Problem sets are provided at the end of each chapter, to be solved either analytically or by numerical methods. The solutions to both types of problems are given as separate pdf files or as Mathematica notebooks (there are 88 of them), all together on a CD accompanying the textbook. The presence of such a collection of numerical analyses enriches the main text and is one of the characteristic features of the book. With the many interesting systems discussed, the book will also be a useful reference for researchers and teachers. It provides the reader with a unique, enjoyable and rather complete textbook of Quantum Mechanics, destined to set a new standard for many years to come.
With all its remarkable achievements, Quantum Mechanics as established in the fundamental papers written by Heisenberg, de Broglie, Schro¨dinger, Dirac, etc. contains some inconsistencies and internal contradictions. Quantum mechanics deals with quantum processes in which the number and the type of particles do not vary during their interaction, but only their energy and momentum states. The formalism of Quantum Mechanics is not capable of describing neither the variation of the number of particles, nor the transformation of one type of particle into another type. Even if in some cases the number of particles is allowed to change, this is somewhat forced and inconsistent. For example, Quantum Mechanics is not able to give a satisfactory explanation of the spontaneous emission, in which an electron situated on an excited energy level emits a quanta γ and decays to a lower energy level.
Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.