We solve the inverse problem corresponding to the fundamental problem of the classical dynamics of a material particle through a matrix treatment: assuming knowing the mass and the position (the trajectory, in relation to an inertial reference) of a particle at all times, we impose that this corresponds to the eigenvector of a “position matrix". Subsequent development leads to a “force matrix", which has the resultant force on the particle as its eigenvector. We identified some limitations of this matrix treatment.
We solve the inverse problem corresponding to the fundamental problem of the classical dynamics of a material particle through a matrix treatment: assuming knowing the mass and the position (the trajectory, in relation to an inertial reference) of a particle at all times, we impose that this corresponds to the eigenvector of a “position matrix". Subsequent development leads to a “force matrix", which has the resultant force on the particle as its eigenvector. We identified some limitations of this matrix treatment.
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.
Concept of inertial mass is not well defined in physics. For defining inertial mass of a particle we need to know its acceleration under some force in an inertial reference frame, which itself is defined by the motion of its background objects. Therefore, the inertial mass of a particle is not completely its intrinsic property and depends on the position of the particle itself and and all other particles (which we refer as background) in the universe. But the background of a particle keeps on fluctuating randomly due to different physical phenomenon in the universe. Therefore, the exact position or the mass of a particle can not be determined at any time, without having full information about all the other particles in the universe. Hence, in this paper, we define the dynamics of a particle statistically. We show that the fluctuations in the background of an object contributes to an extra energy term in the total energy of the particle. If we treat this extra energy term as the quantum potential then it leads to the Schrodinger equation. Some examples are also given to show how a classical interpretation can be given to some quantum mechanical phenomenon.
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 problem of the motion of a particle in an inverse-cube central force field is fully treated by quantum mechanics and the results compared with the classical theory. Taking the effective radial potential energy as r2S, although the solutions for negative energy for $0\ensuremath{\geqq}S\ensuremath{\geqq}\frac{\ensuremath{-}{h}^{2}}{32{\ensuremath{\pi}}^{2}\ensuremath{\mu}}$ satisfy the usual boundary conditions, they can not be admitted because the Hamiltonian is not Hermitian in these solutions. This corresponds to taking (l+21)2 in place of l(l+1) as the analogue of the square of the classical angular momentum. If we do this, we get a complete analogy between the classical and quantum mechanically allowed solutions, with no quantization. The solutions involve Bessel functions of both real and imaginary orders with both real and imaginary arguments.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.
Concept of inertial mass is not well defined in physics. For defining inertial mass of a particle we need to know its acceleration under some force in an inertial reference frame, which itself is defined by the motion of its background objects. Therefore, the inertial mass of a particle is not completely its intrinsic property and depends on the position of the particle itself and and all other particles (which we refer as background) in the universe. But the background of a particle keeps on fluctuating randomly due to different physical phenomenon in the universe. Therefore, the exact position or the mass of a particle can not be determined at any time, without having full information about all the other particles in the universe. Hence, in this paper, we define the dynamics of a particle statistically. We show that the fluctuations in the background of an object contributes to an extra energy term in the total energy of the particle. If we treat this extra energy term as the quantum potential then it leads to the Schrodinger equation. Some examples are also given to show how a classical interpretation can be given to some quantum mechanical phenomenon.