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Two-Dimensional Quantum Mechanics

Published Apr 13, 2006• 0 citations
Publisher

Abstract

Abstract One-dimensional (1D) systems provide examples of many of the most import- ant features of quantum mechanics, but it is also instructive to consider two-dimensional (hereafter 2D or planar) systems for several reasons:Systems with two spatial degrees of freedom provide more opportunities to study multivariable probability concepts, separation of coordinates tech- niques, and new mathematical methods and special functions. They also allow for the visualization of many quantum phenomena which arise in more real- istic three-dimensional (3D) systems, but which are obviously difficult to plot in 3D.

DOI
10.1093/oso/9780198530978.003.0015

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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
O NCE WE leave the realm of either one dimensional or single particle systems, visualizing quantum wavefunctions becomes more challenging. Suppose, for example, we choose to model a single particle confined to a two-dimensional surface? This is analogous to the 1D systems we encountered in Chapter 3 except that the particle can now move in more than one dimension. The wavefunction then has to depend on both x and y . The trouble is that when we attempt to visualize such a wavefunction we can no longer represent the complex value as a 2D phasor, since after using the x and y direction for the position in space, we only have the z direction left for visualization! Since a phase has two independent parts, either (real, imaginary) or (magnitude, phase). We need some way to visualize these two components distributed over some area. One approach is to use color to represent phase, and height in the z direction to represent magnitude. Another would be to place small objects at a distribution of spatial points and use the properties (e.g., color, size, orientation) to represent various aspects of the wavefunction. In this chapter we'll explore a combination of these approaches and develop some insights from the results.
Abstract Dimensionality is a central concept in developing the theory of low-dimensional physics. However, previous research on dimensional crossover in the context of a Bose–Einstein condensate (BEC) has focused on the single-component BEC. To our best knowledge, further consideration of the two-component internal degrees of freedom on the effects of dimensional crossover is still lacking. In this work, we are motivated to investigate the dimensional crossover in a three-dimensional (3D) Rabi-coupled two-component BEC. The spin degrees of freedom consist of the Rabi-like and inter- and intra- interaction coupling constants. The dimensional crossovers from 3D to 2D or 1D are controlled by the continuous increase of 1D or 2D lattice depth respectively. Then we analyze how the dimensionality of the model system combined with spin degrees of freedom can affect quantum fluctuations. Accordingly, the analytical expressions of the ground-state energy and quantum depletion of the system are obtained. Our results show that the dimensional crossover induces a characteristic 3D to quasi-2D or 1D crossover in the behavior of quantum fluctuations, with an emphasis on the separated effects of Rabi-like and inter- and intra- interaction coupling constants on the quantum fluctuations. Conditions for possible experimental realization of our scenario are also discussed.
Abstract Modeling the hydrogen atom is often considered an essential milestone in an introductory quantum mechanics course. Mathematically, the instructional path is clear: from the square potential well, through the harmonic potential, the three-dimensional box, and the spherical well potential prior to the 1/r Coulomb potential. However, there are few experimental examples to help visualize and ground the key physical concepts. Here, we present an experiment where the light scattering of an evaporating, optically levitating water droplet can be used as an analogy to a quantum spherical well potential. We have distilled two important concepts in this analogy: angular momentum and tunneling. This approach is akin to the analogy between a square well potential and a Fabry–Pérot cavity. It can be used as a visual example on the path toward the hydrogen atom in undergraduate-level quantum mechanics courses.