The nature and its changes: Momentum (1st law): p(t)

The continuity of classical mechanics has been the guiding tool for much of the scientific world. The rise of the modern, discontinuous epoch, thanks to Planck’s “key”, observed and ventured the further depths of the physical world. A careful, though open-minded, observation of the possible infinitesimal interactions of the particle-dimension. To understand the movement from classical to quantum physics, we need to understand the tenets of classical mechanics, and subsequent quantum mechanics born from the failure of classical mechanics to account for amany question.  Thus we seek to question the efficacy of quantum mechanics with its multiple interpretations.The question of the validity is critical for science because the whole subject of science depends on accurate methodology and clear objectives. It is essential that the sanctity of scientific knowledge be preserved even though the ongoing life of this subject carries with it corrections and improvements to existing concepts. With each new improvement, the whole system is augmented in terms of the tools made available and knowledge renewed so as to keep it accurate as possible. To begin seeing the benefits of the new science, we need to know its antecedent-classical mechanics.The problem that classical and quantum mechanics attempt to solve is determining how dynamic systems change after an action on it in time. Classical mechanics, which is primarily Newtonian, is for the macroscopic world and by contrast, quantum mechanics microscopic. Now we need to understand the properties of the macroscopic world to see which are static or not, for example, mass. That is why Newtonian mechanics are useful in this regard as we can isolate the changing factors in the system and through that determine how much change has occurred in the system. These changes of the system are described according to the dynamic variables from the initial system in time. So these changes must produce ‘equations of change’, or motion, which will mathematically reflect how the dynamic variables of the system have moved from the initial state in time greater than zero. Classical mechanics assumes that energy is continuous and Newtonian mechanics is the dominant formal structure of classical mechanics. Newtonian mechanics produced two linear equations of which energy was, in a sense, an afterthought derived from momentum and force. Both these laws were elemental in observing nature and its changes: Momentum (1st law): p(t) = mv(t) = mdx(t)/dt; Force(2nd law): F(t) = ma = md2x(t)/dt2. So according to Newtonian mechanics, the state of a point system in time t is in terms of a numerical value of its position x(t) and velocity v(t), that is the rate of change with respect to time, or from the equations above v(t) = dx(t)/dt. Also the other dynamics properties like momentum, kinetic and potential  (thus total) energy depend on the position (x) and velocity (vx) of the point system. The essential key in classical Newtonian mechanics is that the action on the point system is in terms of a force (F) acting on the point system, and this force is proportional to acceleration, a = d2t/dt2, and the constant being mass. So this means that as the force acting on the point system is known, the acceleration, or the second derivative of x(t), would be specified using the equation of Force = ma = md2x/dt2. By integration of the known acceleration, the value of change of v(t) at all times will be known, and the by further integration, the value of x(t) will show what happens to the point system at all times. Therefore, if the position x and the velocity vx of the initial conditions are known and the force acting on it is also known, one can efficiently predict the state of the point system at all times. The key thing is to identify the force, or the 2nd derivative of x with respect to t, because of the specifications of x and vx that specify the change in these two factors, a common concept in Newtonian calculus. So the validity of these principles is verifiable through direct measurements and experimentation in the macroscopic world, therefore, in theory, the complexity of dynamics in the point system is not as significant since we can predict most outcomes. As the principles of classical Newtonian mechanics are essentially the laws of nature in the macroscopic world, we now must consider the microscopic world that is quantum mechanics. These mechanics are essentially involved in the laws of nature that govern the movement in the atomic scale, and assume discrete energy values. Classical Newtonian mechanics predict the outcome for any system with initial conditions known without necessarily having deviations making it very deterministic. Yet with quantum mechanics, this is the opposite case: the known outcome is mostly probabilistic with no manner to be absolutely certain of the result. Also, high attention is needed to quantify the properties of the systems of particles since the atomic level requires many orders of magnitudes of accuracy than those of the macroscopic world rendering absolute predictions very difficult, thus resorting to statistical averages. Without direct measurement for experimental evidence, it becomes difficult to specify the state of a particle in time since the parameters like position and velocity cannot be specified like in the Newtonian world, and there were experimental anomalies that had to be accounted for in electrodynamics in the atomic level. Quantum mechanics is characterized by what is called a “state function”, unlike classical mechanics that is characterized by dynamic variables with specific numerical values. A “state function” is a function of a set of specific variables that are “canonical” to the system in consideration. For example, the state function of a particle moving in linear space along the x-axis is (x), where is the state function of canonical function x. So the state of the particle is described by the functional dependence of (x) on the canonical function x, which is the possible position of the particle. How the state of the particle is described with the changes with time is by (x, t). Incidentally (x, t) also refers to the wave function of the particle as it has similar properties to those of a wave. The state function can also be expressed as a function of another variable which is the linear momentum of the particle p, or (p,t). The question now becomes a matter of formulation where it is by equivalence or representation. When the former state function (x, t) is used, it is in the Schrodinger representation. However, when the latter is used (p, t), it is in the momentum representation. The fact that the same state function is expressible as a function of different variables corresponding to different representation is analogous to electromagnetic theory where a time-dependent electrical signal  can be expressed as either a function of time or its angular frequency. These representations both lead to the similar results for experimental observables of the system, thus the interpretation of these experimental observables should not be differentiated due to the difference of representation. So the use of either representation should be based on mathematical expediency or context. The probabilistic nature of the measuring process in quantum mechanics is embedded in the physical interpretation of the state function. The wave function (x, t) expresses the complex function of x and t, so the wave magnitude of this function gives statistical data on the measurement of the position of the particle. That is to say, in quantum mechanics, the “particle” refers to a statistical “pocket” of particles in the same state. For example, (x, t)2dx would be understood as the probability of finding a particle in the spatial range of x to x + dx in time t in the pocket. So this language of statistical pockets means that we cannot speak of the exact location of a particle in quantum mechanics, unlike in classical Newtonian mechanics, but rather as a statistical average. One could go with how the equation of motion is specified in classical Newtonian mechanics, yet in quantum mechanics action on the dynamic system is generally specified by an observable property corresponding to an energy operator as a function of the position of the system. The physical interpretation of the phase of the wave function imbues the particle with the duality of wave properties, though the statistical interpretation of the wave-particle duality and the measurement process of the dynamic systems present fundamental differences between Newtonian and quantum descriptions of dynamic systems. Inasmuch quantum mechanics is mathematically complete, the central question, however, becomes how quantum mechanics describes the physical world. These mechanics describe how the way microscopic world bears upon the macroscopic, or its affects on measuring instruments, in the language of classical Newtonian mechanics. In theory, quantum mechanics governs the evolution of waves through space-time, but taking these waves as physical entities is problematic in two ways: first, each electron is detected as a particle with a specific location than as a broad wave. It was Born who stated that the intensity of the quantum wave provides the probability that the particle is at a location. Secondly, quantum waves propagate through n dimensions, where n is the number of particles in the system. Hence the standard nomenclature of “wave function” rather than wave as the mathematical function represents a system that has the form of a wave than an actual wave. It was Bohr and Heisenberg that had the first interpretation of quantum mechanics of which it was named Copenhagen Interpretation. Bohr advocated that the conception of the world is necessarily classical; the world is thought of in terms definite objects moving through three-dimensional space. Yet quantum mechanics does not allow this conceptualization, both in terms of waves and particles. Quantum mechanics is an effective tool for predicting measurement results that computes the measuring apparatus as input and produces the probabilities for the measurement results as output. This means that quantum mechanics should not be considered as a description of the quantum world nor the change of the quantum state over time as a causal explanation of the observables. However, the Copenhagen interpretation has two impediments: first, Bohr’s insistence that quantum mechanics is not descriptive pushes off  earlier evidence that quantum mechanics comes from its capability to explain results of interference experiments involving particles. Secondly, Bohr’s position requires a cut between the macroscopic world of classical physics and the microscopic world under quantum mechanics. There can be no such “cut” since the macroscopic is made out the microscopic components, meaning that the macroscopic world must obey quantum mechanics. There are other interpretations that present the quantum mechanics in various forms. One sees quantum mechanics as a description of the system; another sees as a partial description; and another as a complete description of the system with incomplete laws of the dynamical system of quantum mechanics which needs supplementing. All in all, all these interpretations present us with different ontological pictures that have each unresolved difficulties. Nonetheless, we can ascribe considerable insight to Heisenberg and Schrodinger for their formulations as the mathematical nature of observables, and the deterministic description of the states through position and momentum operators, respectively.


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