The new approach for the classical mechanics in mordern physical languages is studied in this article. First of all the Newton's laws of motions are restated in the terminology of the quantum mechanics. Bohr's relation takes a fundamental role in this trial and the concept of energy is the first thing considered. Then the other physical quantities such as the velocity, the momentum, mass, and, etc. are the secondary quantities reduced from the concept of energy. As for the concept of particle anti-particle the third law of the Newtonian mechanics needs it and the usual action and reaction are related to the concept of particle and anti-particle, respectively. The part II treats the motion in the phase space or the trajectories in the phase space. If there are the motions of the original particles, then we can associate it the motions of the antiparticles.|この論文では古典力学に対する現代物理学の新しい研究方法が研究されている。第1にニュートンの運動の法則が量子力学のことばで再記述される。この試みでボーアの関係が根本的役割をはたし,エネルギー概念が最初に考察されるべきことがらとなる。ついで,速度,運動量,質量他はこのエネルギー概念から派生する第2義的量であることを示す。粒子-反粒子概念についてはニュートン力学の第3法則がそれを必要とし,通常の作用-反作用がそれぞれ粒子,反粒子の概念に結びつく。第2部は位相空間での運動や位相空間内のトラジェクトリーをとりあつかう。あるものとの粒子の運動があれば,その運動に反粒子の運動を付随させられる。
The new approach for the classical mechanics in mordern physical languages is studied in this article. First of all the Newton's laws of motions are restated in the terminology of the quantum mechanics. Bohr's relation takes a fundamental role in this trial and the concept of energy is the first thing considered. Then the other physical quantities such as the velocity, the momentum, mass, and, etc. are the secondary quantities reduced from the concept of energy. As for the concept of particle anti-particle the third law of the Newtonian mechanics needs it and the usual action and reaction are related to the concept of particle and anti-particle, respectively. The part II treats the motion in the phase space or the trajectories in the phase space. If there are the motions of the original particles, then we can associate it the motions of the antiparticles.|この論文では古典力学に対する現代物理学の新しい研究方法が研究されている。第1にニュートンの運動の法則が量子力学のことばで再記述される。この試みでボーアの関係が根本的役割をはたし,エネルギー概念が最初に考察されるべきことがらとなる。ついで,速度,運動量,質量他はこのエネルギー概念から派生する第2義的量であることを示す。粒子-反粒子概念についてはニュートン力学の第3法則がそれを必要とし,通常の作用-反作用がそれぞれ粒子,反粒子の概念に結びつく。第2部は位相空間での運動や位相空間内のトラジェクトリーをとりあつかう。あるものとの粒子の運動があれば,その運動に反粒子の運動を付随させられる。
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.
In order to preserve the nature of physical laws in different reference frames there are four dimensions: three of space and one of time. These are combined into a single entity called space–time. Many of the mathematical techniques used in the first part of this book now need to be extended into a four-dimensional space–time. We will see that that actual mathematical concepts are not so difficult, and it largely becomes an exercise in notation. Once the mathematical tools are available, Schrödinger’s equation can be re-visited and a new form of quantum physics discovered which allows for the intrinsic property of electron spin and the existence of anti-matter. Finally, the “action principle” can be introduced. We will see that the action principle pervades all of the equations of motion in physics and ultimately accounts for the quantum behaviour of particles, whether they be sub-atomic particles or large-scale particles. Not only does the action principle apply to particles but also to fields. Applied to the electromagnetic field, we find that Maxwell’s equations arise as a consequence of the action principle. The action principle is perhaps of even more fundamental significance than the law of conservation of energy.
Classical mechanics is introduced in order to lay the foundation for quantum mechanics. Newton's laws are given and used to develop various equations of motion. The modern concept of energy is introduced and used to develop the equation of motion. Generalised coordinates are detailed and used to illustrate the usefulness of the appropriate coordinate system for the problem at hand. Brief introductions to the Lagrangian and Hamiltonian alternatives to Newtonian mechanics are given, primarily to show where it is that the quantum mechanical Hamiltonian comes from. Classical angular momentum mechanics is developed and the parallels between linear and rotational quantities is drawn.
The procedure used to "do physics" in the macroscopic world is familiar: You take an object, start it off with a particular position and velocity, subject it to known forces (say gravity or friction, or both), and follow its trajectory. You then measure the dynamical properties (say position or energy) of that object at a later time and compare those measurements with the prediction using Isaac Newton's laws of motion. Newton's laws directly predict what those quantities should be at that later time, so the comparison is straightforward. However, the microscopic laws of physics, quantum mechanics, aren't so simple. The quantum concepts are so alien and counterintuitive that the language used to describe the mathematics and physics is often ambiguous and confusing. Therefore, for you to \emph{really} understand what is going on, some mathematics must be used. This is my attempt to explain some "weird facts" using a minimum of mathematics, yet eliminating as much ambiguity as possible.
Every vertex of the magic cube of physics discussed so far introduced a major element in the development of physics and the whole of science. Mechanics introduced the basic concept of equations of motion, or time-evolution of a physical system given certain initial condition and the associated idea of determinism, even with its generalization in the form of chaos, the dependence of the outcomes of the temporal evolution on the initial conditions. These concepts taken for many particles, or elements of a physical system, led to the probabilistic description of many particle systems, in the realm of statistical physics and thermodynamics, whereby the averages over time of various observable quantities were related to the ensemble averages, that involve classical, so-called Kolmogorov, probability measures, at least for certain “equilibrium” situations. The second law of thermodynamics introduced the concept of entropy and its probabilistic increase, the first truly emergent, probabilistic law in physics (temperature and pressure, being other emergent quantities, apart from entropy, the measure of disorder of a statistical system). In the context of Newtonian gravity, one gets, at least in retrospect the geometric description of gravity in non-relativistic, Galilean spacetime. Galilean relativity, based on the asymmetric mixing of space and time, with the absolute, universal time, true for all observers, gets generalized in special theory of relativity to observer dependent space and observer dependent time, viewed as “slices” of a new concept of spacetime, with the new concept of causal structure, that distinguish timelike, null and spacelike events. This in turn is made into a full dynamical geometric structure of spacetime with a relativistic description of gravity stemming from curvature of spacetime, and a dynamical causal structure, tied to gravity. In our next story, the story of quantum theory, we will encounter fundamental, intrinsic probabilities, that can interfere, unlike the classical probabilities of statistical thermodynamics, and which are consistent with equations of motion of individual elements, and not, as in statistical thermodynamics, with temporal averages over the time evolution of many elements. This is the big new lesson of quantum theory, that transcends the lessons of classical spacetime physics.
# Least Action in Classical Mechanics: # The Concept of Functional # The Principle of Least Action # Conservation of Energy. Constants of the Motion # Particles Interacting Through an Intermediate Oscillator # Least Action in Quantum Mechanics: # The Lagrangian in Quantum Mechanics # The Calculation of Matrix Elements in the Language of a Lagrangian # The Equations of Motion in Lagrangian Form # Translation to the Ordinary Notation of Quantum Mechanics # The Generalization to Any Action Function # Conservation of Energy. Constants of the Motion # The Role of the Wave Function # Transition Probabilities # Expectation Values for Observables # Application to the Forced Harmonic Oscillator # Particles Interacting Through an Intermediate Oscillator # Space-Time Approach to Non-Relativistic Quantum Mechanics # The Lagrangian in Quantum Mechanics