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Cambridge University Press eBooks

Quantum Mechanics of a Bouncing Ball

Daniel F. Styer

Published Feb 24, 2000• 0 citations
Publisher

Abstract

We started to investigate quantum mechanics by considering only the quantization of magnetic arrows. In our explorations we found out that the magnetic arrow had some funny properties (for example, it was possible that m x did not have a definite value), but at first it seemed that other properties, such as the position of an atom, behaved in the familiar classical way. Eventually (section 9.3) we found that it was also possible to have an atom without a definite value for its position. In this chapter, we investigate what happens when we apply quantum mechanics to a particle's position. Ball bouncing from a floor This chapter will show our framework for quantum mechanics in action, by applying it to the problem of a ball bouncing from a floor. Let us use a very fast ball, such as an electron, so that we can ignore the force of gravity. (We restrict ourselves to an electron that is moving fast on a human scale but slow compared to the speed of light, so that relativistic considerations don't come into play. Also, the magnetic arrow associated with the electron has no effect on the phenomena described in this chapter, so I won't mention it again.) Imagine a source of balls that could send a ball flying in any direction, for example a hot tungsten filament that boils off electrons. Suppose a ball begins at point P , bounces off the floor, and ends up at point Q . (Points P and Q are equally distant from the floor.

DOI
10.1017/cbo9781107050709.015

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We started to investigate quantum mechanics by considering only the quantization of magnetic arrows. In our explorations we found out that the magnetic arrow had some funny properties (for example, it was possible that m x did not have a definite value), but at first it seemed that other properties, such as the position of an atom, behaved in the familiar classical way. Eventually (section 9.3) we found that it was also possible to have an atom without a definite value for its position. In this chapter, we investigate what happens when we apply quantum mechanics to a particle's position. Ball bouncing from a floor This chapter will show our framework for quantum mechanics in action, by applying it to the problem of a ball bouncing from a floor. Let us use a very fast ball, such as an electron, so that we can ignore the force of gravity. (We restrict ourselves to an electron that is moving fast on a human scale but slow compared to the speed of light, so that relativistic considerations don't come into play. Also, the magnetic arrow associated with the electron has no effect on the phenomena described in this chapter, so I won't mention it again.) Imagine a source of balls that could send a ball flying in any direction, for example a hot tungsten filament that boils off electrons. Suppose a ball begins at point P , bounces off the floor, and ends up at point Q . (Points P and Q are equally distant from the floor.
Abstract We review the dynamics of a bouncing ball, both from the classical and the quantum points of view, studying the role of its inertial and gravitational masses. We analyze the behavior of quantum particles of different masses, embedded in a uniform gravitational field. We show that the bouncing ball is a useful didactic example to show that the coupling between rest mass and gravity cannot be avoided at the quantum level and that the equivalence principle must be reformulated in quantum mechanics. Galileo’s experiment of falling bodies can be still satisfied if one considers only the average quantum values of observables.
How shall we approach the principles of quantum mechanics? One way is simply to write them down. In fact I have already done that (in the first paragraph of the Preface), but to do so I had to use words and concepts that you don't yet understand. To develop the necessary understanding I will use a particular physical system as a vehicle to propel our exploration of quantum mechanics. Which system? An obvious choice is the motion of a tossed ball. Unfortunately this system, while simple and familiar in classical mechanics, is a complicated one in quantum mechanics. We will eventually get to the quantum mechanics of a tossed ball (in chapter 14, “Quantum mechanics of a bouncing ball”, page 103), but as the vehicle for developing quantum mechanics I will instead use a system that is simple in quantum mechanics but that is, unfortunately, less familiar in daily life. That system is the magnetic needle in a magnetic field. This chapter describes the classical motion of a magnetic needle so that we will be able to see how its classical and quantal behaviors differ. Magnetic needle in a magnetic field A magnetic needle — like the one found in any woodsman's compass — has a “north pole” and a “south pole”. I will symbolize the magnetic needle by an arrow pointing from its south pole to its north pole.
The path integral formulation of quantum mechanics in the semiclassical or WKB approximation provides a physically intuitive way of relating a classical system to its quantum analog. A fruitful way of studying quantum chaos is based upon applying the Gutzwiller periodic orbit sum rule, a result derived by the path integral method in the WKB approximation. This provides some motivation for learning about path integral techniques. In this paper a pedagogical example of the path integral formalism is presented in the hope of conveying the basic physical and mathematical concepts. The ‘‘quantum bouncer’’ is studied—the quantum version of a particle moving in one dimension above a perfectly reflecting surface while subject to a constant force directed toward the surface. The classical counterpart of this system is a ball bouncing on a floor in a constant gravitational field, collisions with the floor being assumed to be elastic. Path integration is used to derive the energy eigenvalues and eigenfunctions of the quantum bouncer in the WKB or semiclassical approximation. The results are shown to be the same as those obtained by solving the Schrödinger equation in the same approximation.
The objective of this chapter is to go over a few of the basic concepts of quantum mechanics in preparation for a discussion of spectroscopy, which is in many ways the business end of quantum mechanics, at least for chemical scientists. We will also need a few quantum mechanical ideas from time to time in our study of thermodynamics and of kinetics. Why should we learn quantum mechanics at all? Atoms and molecules are small, and their constituent parts, electrons, protons and neutrons, are even smaller. Early in the twentieth century, we learned that small things don't obey the laws of classical mechanics. A different kind of mechanics, quantum mechanics, is required to understand chemistry on a fundamental level. In fact, we need different mechanical theories to treat extremes of both size and speed. Figure 2.1 summarizes the situation. There isn't a sharp cut-off between the various sectors of this diagram. Also note that some of the theories are more general than others. We could in principle use quantum mechanics or general relativity to predict the trajectories of tennis balls, but it just isn't worth the effort, given that classical mechanics works perfectly well in this range of masses and speeds. On the other hand, classical mechanics doesn't give very good results for things that are either extremely large, or small, or fast. Like it or not, to discuss phenomena on an atomic scale, we need quantum mechanics. Ordinary (non-relativistic) quantum mechanics is generally adequate, although electrons in heavy atoms sometimes reach relativistic speeds (approaching the speed of light, c ), requiring relativistic quantum mechanics.
The wave function Before beginning to describe the quantum theory of measurement, we shall discuss briefly the main concepts of quantum mechanics. Readers familiar with quantum mechanics may skip this section and go on to section 2.2. Consider, as a pedagogical aid, a classical particle that can move only along the x direction. For example, it might be a bead on an infinite needle. If we know at any moment of time the values of two numbers, the particle's position and momentum, then we can compute from them all other features of the particle's motion: the forces acting on it, its resulting acceleration, etc. In other words, this pair of numbers, the position and momentum, completely define the state of the classical particle at the chosen moment of time. Similarly, for three-dimensional motion the state is defined by six numbers: the three spatial coordinates x, y, z and the three corresponding components of the momentum p x , p y , p z . A quantum object, with its dual particle-like and wave-like properties, is much more complex than a classical particle. Interferometric experiments (chapter I) show that the object can be spread out over some region of space, or for the bead on a needle, over an interval of x .