The branch of physics referred to as classical mechanics describes and predicts with great accuracy the motion of macroscopic objects moving much slower than the speed of light when subject to one or more forces. At the center of classical mechanics is Newton’s famous second law of motion: force = mass × acceleration. Classical mechanics is deterministic in that it exactly describes (within its limitations) the future motion of an object, provided that the object’s mass, initial position, and velocity, as well as the forces acting on the object, are sufficiently well known. The objects in classical mechanics are often approximated as fictional particles of a certain mass, their sizes being negligible or irrelevant. In cases where the physical extent of an object needs to come into play, the object often takes the form of a sphere or some other calculation-friendly shape (recall, for example, our spherical asteroid from the preface to this book).
The branch of physics referred to as classical mechanics describes and predicts with great accuracy the motion of macroscopic objects moving much slower than the speed of light when subject to one or more forces. At the center of classical mechanics is Newton’s famous second law of motion: force = mass × acceleration. Classical mechanics is deterministic in that it exactly describes (within its limitations) the future motion of an object, provided that the object’s mass, initial position, and velocity, as well as the forces acting on the object, are sufficiently well known. The objects in classical mechanics are often approximated as fictional particles of a certain mass, their sizes being negligible or irrelevant. In cases where the physical extent of an object needs to come into play, the object often takes the form of a sphere or some other calculation-friendly shape (recall, for example, our spherical asteroid from the preface to this book).
The aim of this chapter is to present some very basic principles of classical mechanics. They can help us understand the novelty of quantum theory. Classical mechanics was founded by Newton and contemporary 17th-century natural philosophers, and further developed in the 18th and 19th centuries by many physicists and mathematicians, as a result of the search for a rational, causal, and scientific explanation of the universe. In classical mechanics, real-world objects are modeled as a collection of point particles, i.e., objects with negligible size. The motion of a point particle is characterized by a small number of parameters such as its position, mass, velocity, and the forces applied to it. The position of a point particle is defined with respect to an arbitrary fixed reference point in the three- dimensional Euclidean space, which is usually chosen as the origin of some coordinate system. The velocity of a point particle is the rate of change of its position with time. The force applied to a point particle (called the external force) is the cause for the point particle x y z A to undergo a certain change in its velocity, i.e., an acceleration (Newton’s second law of motion). For the sake of simplicity, here we shall refer to a point particle or a collection of point particles simply as an object. Newton’s first law of motion states that if an object experiences no net external force, then its velocity is constant, i.e., the object is either at rest or it moves in a straight line with a constant speed. The first law of motion postulates the existence of a certain set of frames of reference called inertial reference frames, relative to which the motion of an object not subject to external forces is a straight line at a constant speed. For this reason, Newton’s first law is also referred to as the law of inertia.
The mathematician plays a game in which he himself invents the rules, while the physicist plays a game in which the rules are provided by nature, but as time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which nature has chosen. (Paul A. M. Dirac) As described in Chapter 5, classical ( Newtonian ) mechanics is centered around Hamilton's principle and the associated Newton's second law, particularly as applied to statics and dynamics of nondeformable (discrete) and deformable (continuous) bodies. Such systems are typically of O (1) size operating at O (1) speeds. Modern physics generally refers to branches of physics that have arisen since Einstein's relativity theory in the early part of the twentieth century, including special relativity, general relativity, and quantum mechanics, for example. Modern physics addresses physical phenomena that occur at very small sizes, such as atoms, or very large sizes, such as galaxies, and/or very high speeds close to the speed of light. We focus on the theory of relativity and quantum mechanics, both of which reduce to the classical mechanics limit that has occupied our considerations thus far. Relativity reduces to classical mechanics when υ ≪ c , where c is the speed of light in a vacuum, and υ is a characteristic velocity of the system; quantum mechanics reduces to classical mechanics when length scales dominant within the system are much larger than Planck's universal constant ħ.
Newton's second law of motion, F = ma, has long been considered a cornerstone of classical mechanics, providing a fundamental framework for understanding the behavior of macroscopic objects. However, as physics delves into the quantum realm, the applicability of Newton's second law becomes limited, and the principles of quantum mechanics emerge as the dominant description of particle behavior. Beginning with an overview of the historical development of classical mechanics by Newton and the emergence of quantum mechanics in the early 20th century, the article delves into the foundational principles of framework. The classical description of particles as point masses with deterministic trajectories is contrasted with the probabilistic nature of quantum particles described by wave functions and superposition principles. Practical applications and implications of these theories are also discussed, shedding light on their significance in various scientific disciplines. This research article presents a comparative analysis of Newton's second law of motion and quantum physics, examining their fundamental principles and implications through the lens of classical and quantum mechanics. Beginning with an overview of Newton's second law and its significance in classical mechanics, the article delves into the foundational postulates of quantum mechanics and their departure from classical concepts. This explores the inherent limitations of Newton's second law when applied to quantum phenomena, comparing and contrasting its principles with those of quantum physics. Through a comprehensive analysis of evidence and theoretical frameworks, this article elucidates the boundaries of classical mechanics and highlights the need for quantum mechanics to describe phenomena at the atomic and subatomic levels. Through an exploration of key differences and similarities, this aims to provide insights into the behavior of particles at both macroscopic and microscopic scales
The following equation pair shows the close connection between quantum and classical mechanics. F = m × a = ×( ) ( )average F m average a The first equation is the basis of classical mechanics. It summarizes Newton’s laws, where F is the force, m is the mass, and a is the acceleration. The second equation is the quantum-mechanical generalization. It looks almost the same except for the word average. Averaging is necessary because particles in quantum mechanics are associated with waves, and one cannot precisely determine the position or motion of a wave. Only the averages are known.
In this note we argue there exist two force delivery schemes which are already well-known in classical mechanics namely: F(x)dx and F(t)dt. The former is associated with work and kinetic energy, is deterministic (yields x(t)) and requires information about velocity (or rest mass). It deals with F(x) where F=Force. The second is associated with an impulse which changes momentum. In this case, one does not need to know the rest mass or velocity. For example, a particle with momentum p (this allows for many mo,v pairs with the same p) strikes a surface and is reflected back with -p. It is also associated with probability as such a scenario exists for light reflecting and refracting from a surface. One does not know a priori the probabilities for these two phenomena so the problem is probabilistic. Classical mechanics primarily deals with the F(x) force delivery scheme, e.g. solving a differential equation(s) which arise directly from dp/dt = F(x) or using Lagrange’s or Hamilton’s formalism. Ultimately one obtains the deterministic result x(t). Although the notion of an impulse is well-known in Newtonian mechanics, it does not seem to be used much in calculations because an explicit form of F(t) is usually not known i.e. the force acts in a very tiny dx region in a very short period of time. As noted above, both schemes i.e. both deterministic and probabilistic situations appear in the macroscopic classical world, yet for some reason it is suggested that Newtonian mechanics is only deterministic. Newton considered a photon to be a particle with momentum p. A single photon reaching a surface with a different index of refraction has a certain probability to reflect and another to refract. This is a probabilistic, not deterministic, scenario governed by an impulse delivery. The probabilistic problem of light reflecting and refracting in a macroscopic situation may be solved by using continuity of the periodic electric field and its derivative from Maxwell’s solution for a photon from his electromagnetic equations. We have suggested in previous notes, however, that no knowledge of Maxwell’s result is needed. A steady state reflection/refraction problem removes time (unlike the x(t) deterministic picture), but seems to require some kind of balance on the right and left hand sides of the medium. Furthermore two equations are needed to solve for the two unknowns (i.e. reflection and refraction probabilities). Intensity/flux does not balance because the incident and reflected rays have positive flux. For n1<n2, it would be good to have a balance function which allows partial cancellation of the incident and reflected fluxes. The solution exp(ipx) allows exactly this and this solution only requires knowledge of p, momentum, not of rest mass (i.e. velocity). Thus there seem to be two force delivery mechanisms in the macroscopic classical world with very different solutions. The first is the F(x) delivery yielding the deterministic x(t) and requiring knowledge of mo (rest mass). The second is the impulse delivery which is probabilistic, makes use of exp(ipx), and does not require rest mass. If rest mass is known, however, exp(ipx) is linked with the average equation x = p/m t. At first the two delivery schemes seem mutually exclusive, but in the impulse situation, we try to show that a series of impulses may create the semblance of an F(x). Thus one has the exp(ipx) from the probabilistic force delivery scheme, but at the same time an F(x) emerges and so an x(t) type solution should also follow. How can this be? exp(ipx) is probabilistic and one may create averages with probabilities. Such an average or root mean square average may exhibit deterministic behavioiur. Thus exp(ipx) may be used together with a potential V(x) to create a solution W(x)= Sum over p a(p)exp(ipx) such that the prms(x) behaves like a Force(x) delivery result, yet at the same time one has the features of the impulse delivery as exp(ipx)s are used. Thus the exp(ipx) impulse approach may be used in strict impulse delivery problems and in F(x) delivery problems. This is exactly what occurs in quantum mechanics.
This chapter begins by explaining why physics is the fundamental of all natural sciences. Throughout most its history, atoms were hardly the center of attention in physics. Once called natural philosophy, physics originated in the study of the motion of inanimate objects. From those studies emerged a body of knowledge, including various natural laws, that today we call ‘classical’ physics. Newton's laws of motion, the basis of the science of mechanics, come to mind, as does his law of universal gravitation. These laws are obeyed by objects that in some sense are ‘large’ and, in fact, the laws were discovered after careful observation of the behavior of such objects. The discussion then turns to the focus of the book—the quantum world—where the laws of physics turn out to be different from the laws of classical physics used to explain phenomena in the macroscopic world of everyday life. The book discusses a number of mostly recent experiments on quantum mechanical foundations, most of them involving light at the level of a few photons—sometimes only one.