Physics
Classical Mechanics vs Quantum Mechanics
Classical mechanics describes the motion of macroscopic objects using Newton's laws of motion and is deterministic in nature. Quantum mechanics, on the other hand, deals with the behavior of particles at the atomic and subatomic levels, where particles can exist in multiple states simultaneously and are described by wave functions. Quantum mechanics introduces uncertainty and probabilistic outcomes, unlike classical mechanics.
Written by Perlego with AI-assistance
Related key terms
1 of 5
11 Key excerpts on "Classical Mechanics vs Quantum Mechanics"
eBook - PDFQuantum Dynamics
Applications in Biological and Materials Systems
- Eric R. Bittner(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
1 Survey of Classical Mechanics Quantum mechanics is in many ways the cumulation of many hundreds of years of work and thought about how mechanical things move and behave. Since ancient times, scientists have wondered about the structure of matter and have tried to develop a generalized and underlying theory that governs how matter moves at all length scales. For ordinary objects, the rules of motion are very simple. By ordinary, I mean objects that are more or less on the same length and mass scale as you and I, say (conservatively) 10 − 7 m to 10 6 m and 10 − 25 g to 10 8 g moving at less than 20% of the speed of light. On other words, almost everything you can see and touch and hold obeys what are called classical laws of motion. The term classical means that that the basic principles of this class of motion have their foundation in antiquity. Classical mechanics is an extremely well-developed area of physics. While you may think that because classical mechanics has been studied extensively for hundreds of years there really is little new development in this field, it remains a vital and extremely active area of research. Why? Because the majority of universe “lives” in a dimensional realm where classical mechanics is extremely valid. Classical mechanics is the workhorse for atomistic simulations of fluids, proteins, and polymers. It provides the basis for understanding chaotic systems. It also provides a useful foundation of many of the concepts in quantum mechanics. Quantum mechanics provides a description of how matter behaves at very small length and mass scales, that is, the realm of atoms, molecules, and below. It has been developed over the past century to explain a series of experiments on atomic systems that could not be explained using purely classical treatments. The advent of quantum mechanics forced us to look beyond the classical theories. However, it was not a drastic and complete departure.
eBook - PDF- Yuri A Berezhnoy(Author)
- 2005(Publication Date)
- WSPC(Publisher)
Chapter 1 Quantum Mechanics 1.1 Why Two Types of Mechanics? We live in a complicated world. Our sense organs provide a steady flow of information regarding the numerous phenomena that surround us. Powerful technological inventions also extend the reach of the human senses, giving us access to information more exact and complete than would otherwise be available. The world of our sense perception is the macroscopic one. Here phys-ical phenomena are described by classical physics, which includes classical mechanics, continuum mechanics (hydrodynamics and the theory of elastic-ity), thermodynamics, and electrodynamics. Because classical physics deals with phenomena in which microscopic structure plays no significant role, it cannot yield a comprehensive theory of the structure of real substances. The laws of classical physics govern the motions of objects whose linear dimensions are sufficiently large: R c > 10 6 m, say. Nothing more powerful than an optical microscope will be needed to observe such objects. Classical mechanics, in particular, describes the motions of planets, comets, stars, and galaxies. But there exists another world, inaccessible to direct observation through our sense organs. This is the amazing world of micro-objects, in which physical phenomena are subject to the laws of quantum mechan-ics. The dimensions of molecules, atoms, atomic nuclei, and elementary particles are very small and could be characterized as -R qu < 1(T 8 m. l 2 The Quantum World of Nuclear Physics Thus we have two distinct physical theories. One describes macroscopic phenomena, the other microscopic phenomena. Why do two types of me-chanics exist? The answer is far from simple. Let us pursue the question in more detail. Until the end of the 19th century, practically all physical phenomena were described using classical mechanics. This subject was originally ex-pounded by Sir Isaac Newton in his Philosophiae naturalis principia math-ematica, a monumental work published in 1687.
eBook - PDF- Emmanuel Haven, Andrei Khrennikov(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
Part I Physics concepts in social science? A discussion 1 Classical, statistical, and quantum mechanics: all in one This chapter provides for a very short course on classical (Newtonian as well as statistical) mechanics and quantum mechanics. Readers who have not been trained in physics will be able to gain basic knowledge of the main physical theories developed during the last 400 years, with the inclusion of some of the interpretational problems of these theories. 1.1 Newtonian mechanics We discuss one of Newton’s laws, namely Newton’s second law: “the product of mass and acceleration is equal to the force” or in mathematical symbols: ma = f. (1.1) We state that m is the mass of a particle, a is its acceleration, and f is the force acting on the particle. Newton also introduced the notion of a continuous (infinitely divisible) physical space which was used to describe the dynamics of a particle. Here we can also men- tion the contribution of Leibniz. However, the rigorous mathematical formalization of the real continuum was done much later, at the end of the nineteenth century. Physical space was represented by the mathematical model R 3 , the Cartesian prod- uct space R × R × R of three real lines. In this mathematical model, Newton’s second law can be formalized in the following way. Let us introduce the following notations. Let q = (x, y, z), be a three-dimensional vector, where x, y, z are the particle’s coordinates: v = dq dt = dx dt , dy dt , dz dt (1.2) 3 4 Classical, statistical, and quantum mechanics: all in one is the vector of velocity, and, finally: a = dv dt = d 2 x dt 2 , d 2 y dt 2 , d 2 z dt 2 (1.3) is the vector of acceleration. The dynamics of a particle, t → q (t ) (its trajectory in physical space), is described by the ordinary differential equation: m d 2 q (t ) dt 2 = f (t, q (t )).
eBook - PDF- Nelson Bolívar(Author)
- 2020(Publication Date)
- Arcler Press(Publisher)
Fundamentals of Physics 3 i.e., the so-called thermal death of the universe (Reimann and Hänggi, 2002; Schmid and Zieģelmann, 2017). Quantum mechanics is a whole new world of physics which introduces us to non-traditional laws of physics. It seems that quantum mechanics provides to us a universal opinion in which: • There exists a loss of certainty and an unremovable, unavoidable randomness encompasses the physical world. Albert Einstein was extremely dissatisfied with this quantum physical concept, as stated in his well-known saying: “God does not play dice with the universe.” It is also predicted that the method of performing an observation may disturb the focused subject in an irrepressibly random way (even if there is no physical contact of the object with the outside world). • All the physical systems seem to behave as if they are performing a variety of mutually exclusive tasks simultaneously. For example, an electron bombarded at a wall having two holes in it can seem to act as if it passes through both of the holes simultaneously (Popper, 1950; Kofler and Brukner, 2007). • Broadly disjointed physical systems can act as if they are somehow entangled by a ‘spooky action at a distance.’ In this way, these physical systems are interconnected in mysterious ways that seem to defy either the rules of special relativity or the laws of probability (Brandt and Physics, 1973; Boyer, 1984). The third property of quantum mechanics mentioned above leads us to the deduction that there are some facets of the contemporary physical world which are hard to be termed objectively ‘real’ (Averill and Keating, 1981; Schulze et al., 2000). In short, all of the above three points clearly defy our classical standpoint of the world. 1.2. CLASSICAL PHYSICS Before the introduction of quantum mechanics for understanding the affairs of the physical world, it is worthwhile to comprehend the classical laws of physics.
eBook - PDFFundamental Principles of Modern Theoretical Physics
International Series of Monographs in Natural Philosophy
- R. H. Furth, D. Ter Haar(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
PART I Quantum Theory This page intentionally left blank CHAPTER 1 The Fundamental Principles of Classical Physics ALTHOUGH, as explained in the introduction, the fundamental principles of quantum theory are in many respects different from those of classical phy-sics and the quantum laws are formulated differently from the classical laws, quantum theory has, nevertheless, grown organically from classical theory. It was therefore considered to be necessary to give in this chapter first a brief review of the main methods employed in classical physics to an extent to which they will subsequently be needed. 1.1. Discontinuum or particle theory In the discontinuum or particle theories it is assumed that material physical systems 1111 space discontinuously, being built up of particles which, on any observable scale, are so small that their position in space is completely determined by a position vector r with respect to a suitable system of co-ordinates.t The state variables of a closed system are therefore identical with the position vectors r i and the velocity vectors n i = i i of all its constituent particles. The particles are acted upon by forces F i which are partly external, having their origin outside the system, and partly internal, acting between the particles. The task of the theory is to set up equations between the coordin-ates, the forces, and the time which make it possible to determine the motion of the particles at any time when the state of the system at time t = O is given. In Newtonian dynamics the equations of motion have the form m i i i = m i ll i = F i (i = 1, 2, ..., N) (1.1.1) where m i is the (constant) mass of the ith particle, F i the force on this particle, which in general is a function of the position and velocity of that particle and may also depend on the positions and velocities of the other t Throughout this book vectors will always be denoted by symbols in heavy type. 3
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 2007(Publication Date)
- WSPC(Publisher)
Chapter 2 AN OVERVIEW OF CLASSICAL MECHANICS Theprimarymotivationforthischapteristolaythegroundworknecessaryformaking connections between the classical and quantum-mechanical description of a physical system. In particular, the dynamic description of mechanical systems in terms of La-grangian and Hamiltonian mechanics is presented, with a special emphasis on oscilla-tors. It is only by introducing the so-called canonicalcoordinates and by representing the total energy in Hamilton’s form that one can ultimately make the transition from a classical to a quantum mechanical description of both atomic systems and the interacting electromagnetic radiation field. 2.1 The Lagrangian Formulation In many simple mechanical systems, the most natural way to obtain the equations of motion is to apply Newton’s Second Law to each particle, namely ˙ p i = F i , (2.1) which relates the force F i acting on the i th particle in the system to the rate of change * of the linear momentum p i = m i v i of that particle, where v i is the velocity of particle i . In most cases, the mass m i of each particle is fixed and Newton’s Second Law reduces to the more familiar form F i = m a i , (2.2) where a i = ˙ v i is the acceleration of a given particle in the system. Consider, for example, the simple case of a single particle under the influence of a linear restoring force along the x -axis. The equation of motion is that for a simple harmonic oscillator, i.e., m ¨ x = -Kx, (2.3) where K is a positive (spring) constant. Alternatively, this can be written as d dt parenleftbigg 1 2 m ˙ x 2 + 1 2 Kx 2 parenrightbigg =0 , (2.4) * Each dot above a letter indicates a time derivative. 6 An Overview of Classical Mechanics which means that the expression in the parentheses remains fixed (or conserved) as the system evolves in time. This constant of the motion can be defined as E , the total energy of the system: E = 1 2 m ˙ x 2 + 1 2 Kx 2 .
eBook - PDF- Dietrich Marcuse(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
C H A P T E R R E V I E W O F Q U A N T U M M E C H A N I C S 1.1 H a m i l t o n i a n M e c h a n i c s Formulation of Hamiltonian Mechanics The transition from classical mechanics to quantum mechanics can be made most conveniently and easily by using the Hamiltonian formulation of classical mechanics. Before giving a brief outline of quantum mechanics we, therefore, review a few of the most important equations of the classical Hamiltonian mechanics. The methods of classical mechanics and quantum mechanics are vastly different. Classical mechanics is based on the assumption that any physically interesting variable connected with a particle, such as its position, its velocity, or its energy, can be measured with arbitrary precision and without mutual interference from any other such measurement. Classical mechanics, there-fore, uses sets of variables and functions of these variables to enable us to predict the behavior of physical systems by providing us with differential equations that determine the changes of these functions in space and time. Quantum mechanics is based on the realization that the measuring process may affect the physical system. It is, therefore, impossible in principle to measure simultaneously certain pairs of variables with arbitrary precision. The measurement of one variable affects other variables in such a way that it prevents us from knowing what their values might have been. The mathe-matical formulation of the laws of physics that takes this basic idea into account is very different from the mathematical formulation of classical mechanics, as we shall see later in this chapter. The laws of classical mechanics can be expressed in various mathematical forms. The simplest formulation is based upon Newton's law stating that the 1 2 Review of Quantum Mechanics CHAPTER ONE force acting on a body is equal to the product of its mass times its acceleration.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
Present By the end of the 20th century, classical mechanics in physics was no longer an independent theory. Along with classical electromagnetism, it has become imbedded in relativistic quantum mechanics or quantum field theory. It defines the non-relativistic, non-quantum mechanical limit for massive particles. Classical mechanics has also been a source of inspiration for mathematicians. The realization was made that the phase space in classical mechanics admits a natural description as a symplectic manifold (indeed a cotangent bundle in most cases of physical interest), and symplectic topology, which can be thought of as the study of global issues of Hamiltonian mechanics, has been a fertile area of mathematics research since the 1980s. ________________________ WORLD TECHNOLOGIES ________________________ History of Quantum Mechanics Niels Bohr’s 1913 quantum model of the atom , which incorporated an explanation of Johannes Rydberg's 1888 formula, Max Planck’s 1900 quantum hypothesis, i.e. that atomic energy radiators have discrete energy values ( ε = hν ), J. J. Thomson’s 1904 plum pudding model, Albert Einstein’s 1905 light quanta postulate, and Ernest Rutherford's 1907 positive atomic nucleus discovery. The history of quantum mechanics as this interlaces with history of quantum chemistry began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859-1860 winter statement of the black body radiation problem by Gustav Kirchhoff; the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be discrete; and the 1900 quantum hypothesis by Max Planck that any energy radiating atomic system can theoretically be divided into a number of discrete ‘energy elements’ ε (epsilon) such that each of these energy elements is proportional to the frequency ν with which they each individually radiate energy, as defined by the following formula:
- Malin Premaratne, Govind P. Agrawal(Authors)
- 2021(Publication Date)
- Cambridge University Press(Publisher)
2 Quantum-Mechanical Framework Quantum mechanics is not a theory about reality, it is a prescription for making the best possible predictions about the future if we have certain information about the past. G. ’t Hooft, Journ. Stat. Phys. 53, 323 (1988) 2.1 Review of Classical Mechanics A mathematical description of a nanoscale device is typically based on the equations of motion describing how different parts making up the device change with time [70]. Such a description depends heavily on our understanding of the laws governing the device and on the approximations adopted in formulating the equations of motion. For example, a single device could have different equations of motion, depending on whether we use Newtonian mechanics, quantum mechanics, or relativistic mechanics to describe it [71]. So, one may ask whether a single set of equations of motion can be used if one agrees in advance on the physics describing the device. The answer is clearly no, because the equations of motion change with the type of coordinates used for locating various parts of a system relative to an agreed reference point (the origin of the coordinate axes). The only requirement for such coordinates is that they should be sufficiently unique to identify each and every part of a device. 2.1.1 Generalized Coordinates The coordinates used in classical mechanics are referred to as generalized coordinates [72, 73]. As different choices of generalized coordinates result in different equations of motion for the same device, the equations of motion for any device are not unique. The situation is further complicated because specific properties of the components, referred to as the constitutive relations, are required to thread the physical laws governing the constraints restricting motion of the device [74].
eBook - PDF- David Z Albert, David Z. ALBERT(Authors)
- 2009(Publication Date)
- Harvard University Press(Publisher)
▲▲▲ C H A P T E R S E V E N QUANTUM MECHANICS 1. THE BACKGROUND Newtonian mechanics—as I mentioned at the outset—happens not to be the mechanics of our world; the mechanics of our world (insofar as anybody can tell at present) is quantum mechanics. The empirical predictions of those two theories more or less coincide, of course, insofar as the sorts of things that Newtonian mechanics manifestly gets right (things like the motions of planets, or of baseballs, or even— under certain circumstances—of molecules) are concerned, but the funda-mental pictures they present of the space of possible physical states and of the evolutions of those states over time are altogether different from each other. And so a question very naturally comes up as to whether and how the sort of universal Newtonian statistical mechanics we have been working out over the past six chapters can (as it were) be adapted to quantum theory. And the conventional wisdom is that this sort of adaptation is (to begin with) a perfectly accomplishable thing, and that (as a matter of fact, and par-ticularly in light of the relative closeness of the quantum and the Newtonian theories to each other insofar as things like the motions of molecules are concerned) the whole business turns out to be an eminently simple and straightforward matter of translation. It (or rather, what seems to me to be the best available rational reconstruc-tion of it in terms of the cleaner and more precise sort of vocabulary we have been working out over the course of this book) goes like this: To begin with, the dynamical laws that govern the evolutions of the in-stantaneous states of quantum-mechanical systems in time (or rather, the dy-namical laws that are usually taken to govern those evolutions—of which more later) involve only ~rst derivatives. 1 And one of the things that entails is that the instantaneous states of quantum-mechanical systems are invariably also the complete dynamical conditions of those systems.
eBook - PDFThe Road to Maxwell's Demon
Conceptual Foundations of Statistical Mechanics
- Meir Hemmo, Orly R. Shenker(Authors)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
We start out, then, by taking classical Newtonian mechanics to be the fundamental theory of the world. Of course, taking classical Newtonian mechanics as the fundamental theory of the world can be only a working hypothesis and not a serious claim, for, as is well known, classical 40 mechanics has been replaced by quantum mechanics and by the special and general theories of relativity; and it is fair to say that according to current physics, classical Newtonian mechanics is believed to be false. 1 Our decision to take classical mechanics as our working hypothesis may seem puzzling: what is the point, one might wonder, in judging the adequacy and scope of thermodynamics on the basis of a theory which we think is false? The reason is that the main problems encountered in classical statistical mechanics are repeated – with rather small conceptual changes – in the transition to quantum statistical mechanics; but the presentation of these problems is much simpler in the classical case. Of course, the problems in the mechanical theory itself are quite different as one moves from classical mechanics to quantum mechanics; but the problems associated with proceeding from mechanics to statistical mech- anics are very similar, whether one moves from classical mechanics to classical statistical mechanics, or whether one proceeds from quantum mechanics to quantum statistical mechanics. 2 And so it seems to us preferable to carry out the discussion in the context of the theory, which simplifies the presentation of the ideas that we wish to convey. And so, in this book, we set out to examine the relationship between classical statistical mechanics and thermodynamics. The central prob- lem concerning this relationship had already been recognized by Maxwell as early as 1867, when the theory of statistical mechanics was just beginning to crystallize. Maxwell had made his point by way of his thought experiment of the Demon.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
