Physics
Quantum Mechanics in Three Dimensions
Quantum mechanics in three dimensions refers to the application of quantum theory to systems in three-dimensional space. It involves the study of the behavior of particles and waves at the quantum level within a three-dimensional framework. This field of study is essential for understanding the behavior of atoms, molecules, and other fundamental particles in three-dimensional space.
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11 Key excerpts on "Quantum Mechanics in Three Dimensions"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ physics, nuclear chemistry, and nuclear physics. Some fundamental aspects of the theory are still actively studied. Quantum mechanics is essential to understand the behavior of systems at atomic length scales and smaller. For example, if classical mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus, making stable atoms impossible. However, in the natural world the electrons normally remain in an uncertain, non-deterministic smeared (wave–particle wave function) orbital path around or through the nucleus, defying classical electro-magnetism. Quantum mechanics was initially developed to provide a better explanation of the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom was developed as an explanation for the electron's staying in its orbital, which could not be explained by Newton's laws of motion and by Maxwell's laws of classical electromagnetism. Broadly speaking, quantum mechanics incorporates four classes of phenomena for which classical physics cannot account: • The quantization (discretization) of certain physical quantities • wave–particle duality • uncertainty principle • quantum entanglement Mathematical formulations In the mathematically-rigorous formulation of quantum mechanics developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called state vectors). Formally, these reside in a complex separable Hilbert space (variously called the state space or the associated Hilbert space of the system) well defined up to a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space, usually called the complex projective space.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
10 Quantum electrons in atoms, molecules, and materials The justification for the formalism of quantum mechanics developed in the last chapter is the rich variety of observed phenomena and real systems that are explained by quantum mechanics and are completely inexplicable by classical Newtonian mechanics and Maxwellian electromagnetic theory. As one example, once physicists developed the picture of an atom composed of a small positively charged nucleus surrounded by a cloud of electrons, they were left with an unanswerable puzzle. While Newtonian physics could explain electrons orbiting around the nucleus under the influence of the attractive Coulomb force, the electrons would be experiencing a constant centripetal acceleration and Maxwell’s theory would predict that accelerating electrons should radiate electromagnetic radiation, lose energy, and spiral into the nucleus in a fraction of a second. So, classical physics cannot even account for one of the most fundamental truths: the stability of the atom and matter! The quantum solution of the hydrogen atom, presented later in this chapter, not only gave a satisfactory solution to this puzzle, but it predicted with exquisite accuracy the actual spectroscopic data for its electronic energy levels in the hydrogen atom. In this chapter, we will look at several examples of quantum systems and devices, using quantum mechanical concepts. 10.1 Schrödinger’s equation in three dimensions Since the real world is three dimensional, it will be necessary to extend the quantum mechanics concepts developed in the last chapter to three dimensions. In the three- dimensional Schrödinger’s equation, the wave function Ψ(x, y, z, t) follows by general- izing our momentum operator from iħ ∂ ∂x to iħr ! , where r ! ¼ ^ x ∂ ∂x þ ^ y ∂ ∂y þ ^ z ∂ ∂z is the gradient operator.
eBook - PDF- Andrew Zimmerman Jones(Author)
- 2024(Publication Date)
- For Dummies(Publisher)
2 The Fundamentals: Quantum Physics Principles and Theories IN THIS PART . . . Find out what quantum physics says about particle states. Look back at how quantum physics has revolutionized our understanding of electromagnetism and matter. Investigate the central interpretations, contradictions, and debates of quantum physics. CHAPTER 4 Quantum Mechanics: Particle States and Dualities 71 Chapter 4 Quantum Mechanics: Particle States and Dualities I n the field of quantum physics, you tend to look at individual particles and how those particles behave. (You dive a little bit into multiple particle systems in Chapter 15.) At the core of your examination is understanding that quantum physics defines individual particles in terms of their particle states. Suppose that you have a particle that is in a certain state at one point; examining the particle further means you look at the likelihood that the particle will be in certain other states at other points. In this chapter, you find out about quantum states of particles. I start with an explanation of the quantum numbers that physicists use to describe quantum states, and some specific ways those states manifest in physics. Then I return to an idea, introduced in Chapter 3, of wave-particle duality, which is central to quantum physics. Finally, I talk a bit about how quantum physics led physicists to predict and then discover antimatter. IN THIS CHAPTER » Exploring the role of quantum numbers » Learning how spin defines particles » Diving deeper into wave-particle duality » Identifying the nature of antimatter 72 PART 2 The Fundamentals: Quantum Physics Principles and Theories Quantifying by Quantum Numbers Particles in quantum physics are in defined states, and these states are repre- sented by a sequence of numbers. The numbers that quantum physicists use to define a quantum state are called quantum numbers. Niels Bohr first used this approach when he developed the Bohr model of the atom, as described in Chapter 3.
eBook - PDF- Ian D. Lawrie(Author)
- 2012(Publication Date)
- CRC Press(Publisher)
While the mathemati- cal developments that constitute quantum mechanics have been outstandingly successful in describing all manner of observed properties of matter, it is fair to say that the conceptual basis of the theory is still somewhat obscure. I my- self do not properly understand what it is that quantum theory tells us about the nature of the physical world, and by saying this I mean to imply that I do not think anybody else understands it either, though there are respectable scientists who write with confidence on the subject. This need not worry us unduly. There does exist a canon of generally accepted phrases which, if we do not examine them too critically, provide a reliable means of extracting from the mathematics well defined predictions for the outcome of any experiment we can perform (apart, that is, from the difficulty of solving the mathemat- ical equations, which can be very great). I shall generally use these without comment, and readers must choose for themselves whether or not to accept them at face value. This chapter deals with non-relativistic quantum mechanics, and I am go- ing to assume that readers are already familiar with the more elementary 141 142 A Unified Grand Tour of Theoretical Physics aspects of the subject. The following section outlines the reasons why clas- sical mechanics has proved inadequate and reviews the elementary ideas of wave mechanics. Although the chapter is essentially self-contained, readers who have not met this material before are urged to consult a textbook on quantum mechanics for a fuller account. The remaining sections develop the mathematical theory in somewhat more general terms, and this provides a point of departure for the quantum field theories to be studied in later chap- ters. 5.0 Wave Mechanics The observations which led to the quantum theory are often summarized by the notion of particle–wave duality.
eBook - PDF- P R Wallace(Author)
- 1991(Publication Date)
- World Scientific(Publisher)
Quantum theory states that matter does not consist of such particles, but rather that matter can exist in a discrete set of non-localized states of a field-like character, rather like the normal modes of a continuous classical system like a vibrating string or drum. We do not know how to interpret intuitively the field quantity or wave function rjj; we can, however, determine the states of energy of systems. This may be done by solving wave equations similar to those known in the theory of classical fields, since it can be shown that properly bounded solutions of these equations can Books on physics for non-physicists never do justice to Hamilton, who was one of the greatest physicists of the 19th century. Aside from his work on optics, his very elegant formulation of classical mechanics forms the basis for Schrodinger's wave mechanics. Schrodinger's wave equation is constructed from the Hamil-tonian function 14 366 Phyrict: Imagination & Reality only be obtained for certain values of the energy, which occurs as a parameter in the equations. When we spoke of bounded systems in classical physics, we imagined strings clamped at their ends, or drums clamped at their periphery. But an oscillating pendulum is also a bounded system, as is a planetary orbit. They are bounded by constraining forces, so that inadequacy of energy keeps them within spatial limits. Classically, the energy of a particle constrained by a force to vibrate about a fixed point is well known. It consists of kinetic energy |mt> 2 , m being its mass and v its speed, and a potential energy kx 2 . The kinetic energy can also be written as p 2 /2m, where p is its momentum, k is the elastic constant characterizing the restoring force, which is of magnitude kx. The vibration takes place about a state in which x = 0 and p = 0 - the mid-point of the oscillation. Thus, the mean square value of the momentum is p 2 , and that of the displacement x 2 .
eBook - PDFPhases of Matter and their Transitions
Concepts and Principles for Chemists, Physicists, Engineers, and Materials Scientists
- Gijsbertus de With(Author)
- 2023(Publication Date)
- Wiley-VCH(Publisher)
30 3 Quantum Mechanics After reviewing classical particle mechanics (PM), in this chapter we discuss the sec- ond “tool,” quantum mechanics (QM). We first describe the kinematics and the kinetics. Subsequently, we deal with the interpretation and some exactly solvable and approximately solvable problems. 3.1 Quantum Concepts For molecular phenomena, classical PM is in principle inadequate, which led to the discovery of QM. Here, we provide a necessarily compact outline of basic quantum theory only and, similar as for PM and later on thermodynamics, we refrain from comparison with experiment. The basic postulate is that on a molecular level all information for a system, that is, the molecule or molecules at hand, is contained in a “state vector” and that any observable property can be extracted from the “state vector” by applying the appropriate operator [1−3]. This abstract formulation (Sections 3.1.1–3.1.3) will be helpful as background for Chapter 19 and can be skipped by readers more interested in the subsequent more down-to-earth sections on interpretation and some practical examples (Section 3.2 and further). 3.1.1 Fundamental Quantum Kinematics A (quantum) state is an abstract quantity, dependent on the number of degrees of freedom (DoF) d of the molecular system, which can be described by a vector in a d-dimensional (dD) complex vector space H, conventionally addressed as Hilbert space. Such a vector is called a ket and labeled as |…⟩. Abstract implies that these vectors are defined by their formal properties. Let , , … be complex numbers and |a⟩, |b⟩, … be kets. Then, by definition, the product |a⟩ and the sum |c⟩ = |a⟩ + |b⟩ are also vectors in the same space. Adding kets is denoted as superposition of states. Because kets are complex quantities, one needs to define to every ket |a⟩ in a one-to-one way a dual vector, labeled ⟨b| and denoted as bra.
eBook - PDF- Mikio Namiki, Saverio Pascazio, Hiromichi Nakazato(Authors)
- 1998(Publication Date)
- World Scientific(Publisher)
Wheeler's statement logically counters the standpoint (engendered by a classical view) that wave and particle are independently (and exclusively) connected to physical entities. Nevertheless, one may say that the concept 34 Elements of QM of quantum mechanical entity is not yet as clear as the classical one. In this book we shall not enter into this kind of epistemological argument. 2 Formulation of quantum mechanics Quantum mechanics is formulated in terms of states (or wave functions) and observables, together with their physical interpretation, and a dynamical principle. 2.1 Fundamental postulates Wave mechanics (i.e. the primitive form of quantum mechanics) starts from the notion of state. State: A state of motion of a dynamical system with n degrees of freedom at time t is described in terms of the wave function ip(x,t), which depends on x = (xi,x<2, ■ ■•,!„), x representing a point in configuration space: For example, n = 3 for one-particle, n — 6 for two-particle, ... and n = ZN for N-particle systems, respectively, if we neglect any kind of constraints. Notice that we deal with a; as if it belonged to a one-dimensional space. For simplicity, let us call such a state V'-state. Furthermore, if the particles have other quantum numbers, the wave function will depend on them, as well. For a while, however, we shall suppress the dependence on these additional quantum numbers. We usually call 4>(x,t) (depending on x) the position representation of a state.
eBook - ePub- Ijaz A. Rauf(Author)
- 2021(Publication Date)
- CRC Press(Publisher)
CHAPTER 3 An Overview of Quantum MechanicsDOI: 10.1201/9781003206743-3Classical mechanics have been continuously developed since Newton’s time and applied to an ever-widening variety of complex systems, including interaction with matter and the electromagnetic field. The philosophy of minimal action governs classical mechanics. When scientists first observed the behavior of electrons and nuclei, they tried to explain their experimental results in terms of Newtonian classical motions, but, in the end, these attempts failed. We find that these small particles behaved in a way that is not in line with Newton’s equations.The classical measurement theory assumes that an interaction between the system of interest and the measuring device can be made arbitrarily small or accurately compensated. One may talk of an idealized measurement that does not interrupt the system’s properties of interest. Nevertheless, in the case of an atomic phenomenon, the interaction between the system and measuring instrument is not arbitrarily minimal. Nor can the interaction-related disruption be accounted for precisely because it is uncontrollable and unpredictable to some degree. A calculation on one property will also result in irreversible changes in the value previously allocated to another feature. To talk of a microscopic system with exact amounts for all its wealth is meaningless. Therefore, atomic physics laws must be represented in a nonclassical language, which is a symbolic representation of microscopic measurement principles.Let us now consider some experimental data that gave rise to these paradoxes and led to quantum mechanics development. Light exhibiting interference was assumed to be a wave phenomenon; however, later, it was observed that shining light on a metal surface results in electrons’ ejection from the metal surface (photoelectric effect). Considering the photoelectric effect based on the wave nature of light would conclude that the ejected electron’s energy would depend on the intensity of light rather than the frequency observed in experiments.
eBook - PDF- Ashok Das(Author)
- 2012(Publication Date)
- WSPC(Publisher)
Chapter 3 Basics of quantum mechanics In the next few lectures, we will introduce the basic concepts of quan-tum mechanics. However, let us first discuss the reasons for going beyond the classical description of physical systems, which we have discussed in the first chapter. 3.1 Inadequacies of classical mechanics Classical mechanics works well when applied to macroscopic or large systems. However, around the turn of the twentieth century (1900-1920), it was observed that microscopic or small systems behaved very differently from the predictions of classical mechanics. We would, of course, discuss more quantitatively what we mean by microscopic systems. But, for the present, let us understand by a microscopic system, a system of atomic size or smaller and list below various difficulties that one runs into in applying the classical description to microscopic systems. 1. Planetary model. The planetary model of the atom, where elec-trons move in definite orbits around the nucleus, was in serious trouble. According to classical mechanics, a particle in such an orbit is being constantly accelerated. Furthermore, we also know that a classical charged particle, when accelerated, emits radiation. Therefore, an electron going around a nucleus would continuously emit radiation and become less and less energetic. This has the consequence that the radius of the orbit would con-stantly shrink in size, until the electron falls into the nucleus. Thus, according to classical mechanics, the planetary motion in atoms was unstable. 2. Blackbody radiation. The theoretical calculation of the black-body radiation spectrum, which assumes that electromagnetic radiation is a wave and, therefore, can exchange energy in any 61 62 3 Basics of quantum mechanics continuous amount, leads to a result which does not agree with the experimental measurement (curve).
eBook - PDFQuantum Physics
An Introduction
- J Manners(Author)
- 2018(Publication Date)
- CRC Press(Publisher)
You should pay particular attention to the fact that quantum mechanics predicts a continuous range of possible energies for the free particle since there is a widespread misconception that quantum mechanics is all about systems in which energy is ‘quantized’ into discrete energy levels. While it is certainly true that quantum physics originated with Planck’s concept of the quantum of energy, and while it is also true that quantum mechanics provides a natural account of energy quantization, it is certainly not true that quantum mechanics is restricted to cases in which energy 66 is discretely quantized. The free particle is clearly a case in which the Schrodinger equation can be written down and solved (even though we have not done so here), and in which the solutions predict a continuous range of possible total energies. The system can be treated using quantum mechanics, yet its energy is not ‘quantized’. When trying to grasp the essence of quantum mechanics, the fact that we learn about the possible values of the energy of a system by solving the Schrodinger equation and considering the relevant wavefunctions is more significant than the issue of whether or not those possible energy values are quantized. Here is a summary of the main result of this subsection: The fundamental equation of quantum mechanics is Schrodinger’s time- dependent equation.
eBook - PDF- Steven Weinberg(Author)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
3.7 Interpretations of Quantum Mechanics The discussion of probabilities in Section 3.1 was implicitly based on what is called the Copenhagen interpretation of quantum mechanics, formulated under the leadership of Niels Bohr. 15 According to Bohr, 16 “The essentially new 15 N. Bohr , Nature 121, 580 (1928), reprinted in Quantum Theory and Measurement, eds. J. A. Wheeler and W. H. Zurek (Princeton University Press, Princeton, NJ, 1983); Essays 1958–1962 on Atomic Physics and Human Knowledge (Interscience Publishers, New York, 1963). 16 N. Bohr, “Quantum Mechanics and Philosophy – Causality and Complementarity,” in Philosophy in the Mid-Century, ed. R. Klibansky (La Nuova Italia Editrice, Florence, 1958), reprinted in N. Bohr, Essays 1958–1962 on Atomic Physics and Human Knowledge (Interscience Publishers, New York, 1963). 3.7 Interpretations of Quantum Mechanics 87 feature of the analysis of quantum phenomena is . . . the introduction of a fundamental distinction between the measuring apparatus and the objects under investigation. This is a direct consequence of the necessity of accounting for the functions of the measuring apparatus in purely classical terms, excluding in principle any regard to the quantum of action.” As Bohr acknowledged, in the Copenhagen interpretation a measurement changes the state of a system in a way that cannot itself be described by quan- tum mechanics. 17 This can be seen from the interpretive rules of the theory. If we measure an observable represented by an Hermitian operator A, and the system is initially in a normalized superposition ∑ r c r r of orthonor- mal eigenvectors r of A with eigenvalues a r , then the state is supposed to collapse during the measurement to a state in which the observable has a definite one of the values a r , and the probability of finding the value a r is given by what is known as the Born rule, as |c r | 2 .
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