Classical mechanics and electromagnetismThe mechanics of motion formulated by Newton and contemporaries in the 17th century is the foundation of contemporary physics.It is immensely successful, explaining and quantifying a vast array of phenomena, ranging from planetary motion to the flow of fluids.By the latter part of the 19th century the study of electric and magnetic fields and their inclusion in the framework of Newtonian physics appeared near completion.Since motion and electromagnetism are central to the failures of Newtonian physics in the atomic realm, it is useful to review the essential ideas, concepts, and the implicit assumptions of what is now referred to as classical physics.Many of the results presented in this chapter are assumed to be familiar to the reader, but some concepts are more thoroughly presented because of their relevance to quantum mechanics. Newtonian mechanicsFor simplicity initial consideration is limited to a single point particle of inertial mass m.A point particle is treated as infinitesimally small and indivisible.It is possible to formulate Newtonian mechanics in terms of continuous matter distributions, referred to as continuum mechanics, but these will not be considered in this text.In that regard, point particles are often used as an approximation to the constituents of real matter in Newtonian mechanics.A point particle's position therefore coincides with a point in space at every given moment of time, parameterized with t, and can be described mathematically by a vector x from the choice of origin to that point.As the point particle moves its position is given by the time-dependent trajectory x t ( ).An important assumption of Newtonian mechanics is that all observers are equipped with identical meter sticks and synchronized clocks, and they all agree upon the current position at the time t, although the choice of each observer's origin is arbitrary.The Cartesian components of the vector trajectory are denoted x t ( ) i , where the subscript i runs over the dimensions of the space under consideration.In one dimension the subscript is
Classical mechanics and electromagnetismThe mechanics of motion formulated by Newton and contemporaries in the 17th century is the foundation of contemporary physics.It is immensely successful, explaining and quantifying a vast array of phenomena, ranging from planetary motion to the flow of fluids.By the latter part of the 19th century the study of electric and magnetic fields and their inclusion in the framework of Newtonian physics appeared near completion.Since motion and electromagnetism are central to the failures of Newtonian physics in the atomic realm, it is useful to review the essential ideas, concepts, and the implicit assumptions of what is now referred to as classical physics.Many of the results presented in this chapter are assumed to be familiar to the reader, but some concepts are more thoroughly presented because of their relevance to quantum mechanics. Newtonian mechanicsFor simplicity initial consideration is limited to a single point particle of inertial mass m.A point particle is treated as infinitesimally small and indivisible.It is possible to formulate Newtonian mechanics in terms of continuous matter distributions, referred to as continuum mechanics, but these will not be considered in this text.In that regard, point particles are often used as an approximation to the constituents of real matter in Newtonian mechanics.A point particle's position therefore coincides with a point in space at every given moment of time, parameterized with t, and can be described mathematically by a vector x from the choice of origin to that point.As the point particle moves its position is given by the time-dependent trajectory x t ( ).An important assumption of Newtonian mechanics is that all observers are equipped with identical meter sticks and synchronized clocks, and they all agree upon the current position at the time t, although the choice of each observer's origin is arbitrary.The Cartesian components of the vector trajectory are denoted x t ( ) i , where the subscript i runs over the dimensions of the space under consideration.In one dimension the subscript is
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.
Newtonian Mechanics, theory of Elasticity and Fluid Dynamics, together with Maxwellian Newtonian Mechanics, theory of Elasticity and Fluid Dynamics, together with Maxwellian Electrodynamics are basically what constitute Classical Physics. Classical Physics was developed and successfully applied to describe the motions of macroscopic bodies like planetary systems and other every day phenomena. It also described electromagnetic field and its interaction with matter. The development of classical physics was almost complete by the early twentieth century. With the discovery of subatomic particles, radioactivity and X-rays and also the accumulation of data on the spectrum of Black Body Radiation and other spectroscopic studies, it became imperative to invoke new concepts and ideas. Thus Quantum Mechanics, the fundamental theory of the physical world was formulated.
In Newtonian mechanics, a particle is described as an object that is characterized by certain properties. The most important characteristics of a particle are its mass, position, velocity, and acceleration. In this chapter, it is shown how a particle follows a well defined classical trajectory. The main characteristics of the dynamics of particles such as linear and angular momentum, force, energy, moment of inertia, and torque are presented. An understanding of these effects is essential in understanding and appreciating the laws of quantum mechanics. As an example of the Newtonian mechanics, the motion of an electron in electric and magnetic fields experiencing Lorentz force is discussed. This example explains how Thomson discovered the electron in the late nineteenth century.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
Newtonian mechanics is the foundation of Classical Physics. Newton’s mechanics, Thermodynamics, Wave theory of Optics and Maxwell’s electromagnetic theory belong to regime of Classical Physics and can be used to explain a wide range of phenomena in the Universe at macroscopic scale. These theories fail spectacularly when applied to phenomenain the atomic and nuclear regime, for example, proton-atom scattering or the flow of electrons in a semiconductor. Quantum mechanics is the most successful scientific theory that has ever been created and it has completely changed our view of the world. The failure of Classical Physics was highlighted by black body radiation and photoelectric effect. Max Planck and Albert Einstein provided explanations of both phenomena based on quantum hypothesis and are thus considered founders of Quantum Physics. The origin of quantum mechanics goes back to the mid-1920s.It was formulated first as matrix mechanics by Werner Heisenberg, Max Born and Pascual Jordan; then as wave mechanics by Louis de Broglie and Erwin Schrödinger; and later on asquantum statistics of subatomic particles by Fermi-Dirac and Bose-Einstein. Combining relativistic mechanics with quantum mechanics, Dirac formulated his relativistic quantum mechanics during 1930s. Uncertainty Principle is thecornerstone of Quantum Physics. The role of randomness in microscopic physical processes shatters the myth that the universe is deterministic. Quantum world is unpredictable in the classical sense and demolishes the idea of an objective universe. The Copenhagen interpretation remains the quantum mechanical formalism that is currently most widely accepted amongst physicists.Quantum theories support cosmic spirit pervading the cosmos and inter – relationship of individuals in the world society. Quantum philosophy is holistic and is going to revolutionize our world-view.
We have seen in the previous chapter how useful wave mechanics can be in solving physical problems. But wave mechanics has several limitations. It describes physical states by means of wave functions, which are functions of the positions of the particles of the system, but why should we single out position as the fundamental physical observable? For instance, we might want to describe states in terms of probability amplitudes for particles to have certain values of the momentum or energy rather than the position. A more fundamental limitation is that there are attributes of physical systems that cannot be described at all in terms of the positions and momenta of a set of particles. One of these attributes is spin , which will be a chief subject of Chapter 4. Another is the value of the electric or magnetic field at some point in space, treated in Chapter 11. This chapter will describe the principles of quantum mechanics in a formalism which is essentially the “transformation theory” of Dirac, mentioned briefly in Section 1.4. This formalism generalizes both the wave mechanics of Schrödinger and the matrix mechanics of Heisenberg, and is sufficiently comprehensive to apply to any sort of physical system. States The first postulate of quantum mechanics is that physical states can be represented as vectors in a sort of abstract space known as Hilbert space . Before getting into Hilbert space, I need to say a bit about vectors in general. In kindergarten we learn that vectors are quantities with both magnitude and direction. Later, when we study analytic geometry, we learn instead to describe a vector in d dimensions as a string of d numbers, the components of the vector. The latter approach lends itself well to calculation, but in some respects the kindergarten version is better, because it allows us to describe relations among vectors without specifying a coordinate system. For instance, a statement that one vector is parallel to a second vector, or perpendicular to a third, has nothing to do with how we choose our coordinate system. Here we will formulate what we mean by vector spaces in general, and Hilbert space in particular, in a way that is independent of the coordinates we use to describe directions in these spaces.