The exact formulation of the correspondence principle and in particular understanding the quantum‐to‐classical transition remains an open problem in quantum mechanics. In this paper we present our investigation into the quantum‐to‐classical transition of the most trivial of quantum systems — a particle in a box. Whilst it is perhaps surprising, even this example can produce new physical insight into these fundamental problems. With modern fabrication techniques of nano‐mechanical systems we will be able to experimentally investigate these results and directly observe the quantum‐to‐classical transition. This will enable us to build technologies that probe the fundamental questions of quantum mechanics, such as the maximum size of a quantum object.
The exact formulation of the correspondence principle and in particular understanding the quantum‐to‐classical transition remains an open problem in quantum mechanics. In this paper we present our investigation into the quantum‐to‐classical transition of the most trivial of quantum systems — a particle in a box. Whilst it is perhaps surprising, even this example can produce new physical insight into these fundamental problems. With modern fabrication techniques of nano‐mechanical systems we will be able to experimentally investigate these results and directly observe the quantum‐to‐classical transition. This will enable us to build technologies that probe the fundamental questions of quantum mechanics, such as the maximum size of a quantum object.
In this chapter, we introduce the particle in a box. This problem has become a standard in most quantum texts, even though it does not accurately represent too many real physical systems. Instead, it is used to illustrate quantum principles because it is viewed as one of the simplest quantum systems to discuss. Within our approach, however, this problem is a bit more challenging to solve. Nevertheless, it does provide a number of interesting insights, and it also allows us to prove an important result about the existence of bound states in one, two, and three dimensions. Our work here also allows us to answer the question why are some eigenvalues in quantum mechanics quantized, while others are not.
The behavior of classical macroscopic particles – from baseballs to planets – is governed by Newtonian mechanics as we have described in the first chapters of this book. The behavior of very small particles, on the other hand, is governed by an entirely different collection of equations and rules that go under the name of “quantum mechanics” and these particles are referred to as quantum particles. These rules have analogies to classical mechanics, but are fundamentally different and somewhat mysterious as we discussed in Chapter 8. The fundamental principles of quantum mechanics cannot be proven to be correct. They were developed based on the results of experimental observation, by analogy with classical mechanics, and through the consideration of logical implications of the formalisms used. There is no way to derive the fundamental elements of quantum mechanics, any more than one could predict from first principles that Newtonian forces should add as vectors. On the other hand, quantum mechanics is among the most rigorously tested theories in science, and the predictions of quantum mechanics have, so far, always been in agreement with experiment. In this chapter, we will present quantum mechanics as a series of postulates and examine their implications in application to simple one-dimensional systems. In the next chapter we will examine quantum effects in real systems including atoms, molecules, crystals, and artificially engineered nano-structures. Postulates of quantum mechanics QM Postulate 1: A quantum particle is described by a wave function Ψ( x , y , z , t ) which is a function of spatial coordinates and time. All physically determinable quantities can be derived from the wave function Ψ( x , y , z , t ) ; quantities which cannot be found from the wave function are not physically meaningful. Note that some things that one could ask about a quantum particle, for example the exact position and simultaneous exact momentum of a particle, are not derivable from the wave function and are therefore considered indeterminate (or uncertain ). These are things which cannot ever be known about a particle, and, as such, they are meaningless to contemplate.
Quantum mechanics has brought revolutionary changes in the conceptual foundations of physics and continues to shape the modern world. In this book, the author's emphasis is on helping students comprehend the significance of the underlying principles and understand the ways the new concepts were introduced. Including many worked examples and problems, this book will be an invaluable resource for students in physics, chemistry and electrical engineering needing a clear and rigorous introduction to quantum mechanics.
The correspondence principle addresses the connection between classical and quantum physics. The simple statement that quantum mechanics reduces to classical mechanics in the limit where the principal quantum number n approaches infinity, while found in many textbooks, is not true in general. In this article we will give special attention to the notion that the quantum frequency spectrum of a periodic system reduces to the classical spectrum in this limit. Two simple counter-examples—a particle in a cubical box, and a rigid rotator—will show us that the classical result is not always recovered in the limit of large quantum numbers.
There is no dissent regarding the quantum theory success on the small-scale structure of the world but there is equally an open subject of debate about its interpretation. Both quantum mechanics and the theory of relativity are indeed extensions of classical mechanics, that is, they reproduce their results in other appropriate limiting cases. Quantum mechanics provides a new set of laws and a method of description of microscopic systems that has been very successful. The chapter discusses some of the fundamental principles of this highly controversial subject. In quantum physics, it is difficult to uphold a sharp distinction between the object and the observer. The chapter examines why quantum mechanics seems to be so hard for newcomers to learn and understand. The theory of quantum mechanics has led to many applications: laser, the electron microscope and many other technological devices.
Ever since the advent of quantum mechanics, it has been clear that the atoms composing matter do not obey Newton's laws. Instead, their behavior is described by the Schroedinger equation. Surprisingly though, until recently, no clear explanation was given for why everyday objects, which are merely collections of atoms, are observed to obey Newton's laws. It would seem that, if quantum mechanics explains all the properties of atoms accurately, they, too, should obey quantum mechanics. This reasoning led some scientists to believe in a distinct macroscopic, or ``big and complicated,'' world in which quantum mechanics fails and classical mechanics takes over, although there has never been experimental evidence for such a failure. Even those who insisted that Newtonian mechanics would somehow emerge from the underlying quantum mechanics as the system became increasingly macroscopic were hindered by the lack of adequate experimental and theoretical tools. In the last decade, however, this quantum-to-classical transition has become accessible to experimental study and quantitative description, and the resulting insights are the subject of this article.