Fundamentals of Quantum Mechanics: Classical mechanics vs. quantum mechanics
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Abstract
What is quantum mechanics and what does it do? In very general terms, the basic problem that both classical Newtonian mechanics and quantum mechanics seek to address can be stated very simply: if the state of a dynamic system is known initially and something is done to it, how will the state of the system change with time in response? In this chapter, we will give a brief overview of, first, how Newtonian mechanics goes about solving the problem for systems in the macroscopic world and, then, how quantum mechanics does it for systems on the atomic and subatomic scale. We will see qualitatively what the differences and similarities of the two schemes are and what the domain of applicability of each is. 1.1 Brief overview of classical mechanics To answer the question posed above systematically, we must first give a more rigorous formulation of the problem and introduce the special language and terminology (in double quotation marks) that will be used in subsequent discussions. For the macro-scopic world, common sense tells us that, to begin with, we should identify the ''system'' that we are dealing with in terms of a set of ''static properties'' that do not change with time in the context of the problem. For example, the mass of an object might be a static property. The change in the ''state'' of the system is characterized by a set of ''dynamic variables.'' Knowing the initial state of the system means that we can specify the ''initial conditions of these dynamic variables.'' What is done to the system is represented by the ''actions'' on the system. How the state of the system changes under the prescribed actions is then described by how the dynamic variables change with time. This means that there must be an ''equation of motion'' that governs the time-dependence of the state of the system. The mathematical solution of the equation of motion for the dynamic variables of the system will then tell us precisely the state of the system at a later time t > 0; that is to say, everything about what happens to the system after something is done to it. For definiteness, let us start with the simplest possible ''system'': a single particle, or a point system, that is characterized by a single static property, its mass m. We assume that its motion is limited to a one-dimensional linear space (1-D, …