With some simple examples behind us, we are now ready to list the rules of quantum mechanics: that is, the rules that we need. Unfortunately, there are more than the three laws of classical mechanics (Newton's laws) or the four laws of thermodynamics. But the rules we need can be written on one page. The remainder of the chapter illustrates the meaning of some of these rules, using simple wave functions that we have already derived. The best way to understand these rules is by practicing their application. For this reason, more end-of-chapter problems are present here than in any other chapter.
With some simple examples behind us, we are now ready to list the rules of quantum mechanics: that is, the rules that we need. Unfortunately, there are more than the three laws of classical mechanics (Newton's laws) or the four laws of thermodynamics. But the rules we need can be written on one page. The remainder of the chapter illustrates the meaning of some of these rules, using simple wave functions that we have already derived. The best way to understand these rules is by practicing their application. For this reason, more end-of-chapter problems are present here than in any other chapter.
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.
We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the ε j 's and g j 's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics. We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.
This chapter gives a summary of the mathematical formalism used in quantum mechanics, with a short bibliography put directly at its end. It should be seen as a complement, to be used by readers who wish to know more than what has been recalled about the mathematical tools in the other chapters. Some results are given without explicit proofs; they can be found for instance in Chapters II and IV of [1] . Many quantum mechanics textbooks introduce its general formalism in a more complete way, for instance Chapter VII of [2] , Chapter 3 of [3] , or Chapter 2 of [4] . We first summarize the general formalism of quantum mechanics for any physical system (§11.1), with the Dirac notation; we then study how this formalism treats the grouping of several physical systems into one single quantum system (§11.2); finally, we study a few simple special cases (§11.3), for instance a single particle in an external potential, with or without spin; the reader who prefers wave functions to more abstract reasonings in spaces of states may begin with this section. General physical system The general formalism of quantum mechanics applies to all physical systems, whether they contain a single particle, many particles of various sorts, one or several fields, etc. Quantum space of states The physical state of a system at each time is defined in quantum mechanics by a state vector which, in Dirac notation, is written ∣ ψ 〈 – or ∣ ψ(t) 〈 if one wishes to make the time dependence explicit.
In this monograph I present the basic structure of an interpretation of quantum mechanics. Chapter 1 outlines the central ideas behind the interpretation and illustrates them by means of examples: Subsequent chapters fill in the details of the interpretation. But it is important to begin by explaining what I take an interpretation of quantum mechanics to be, and why any further interpretation needs to be offered. After all, quantum mechanics (in some form) is by now both a foundation for much of contemporary physics and a veteran of more than sixty years of intermittent but sometimes intense reflection on its content and meaning. What more can be, or should be, said about the interpretation of this theory? Many physicists believe that no more needs to be said: that there is basically only one way of understanding quantum mechanics, due to Bohr, Heisenberg, Pauli, and others, and that the only remaining interpretative task is that of the physics teacher, who seeks to perfect ways of conveying this understanding to new generations of students. Sir Rudolf Peierls, one of the more lucid and distinguished of these physicists, even objects to the use of the familiar term ‘Copenhagen interpretation’ to refer to the way of understanding quantum mechanics due to Bohr, Heisenberg, Pauli, and others. Because this sounds as if there were several interpretations of quantum mechanics. There is only one. There is only one way in which you can understand quantum mechanics.[…]
Basic scientific theories usually start with a set of hypotheses or “postulates.” There is generally no logical reason, apart from internal consistency, that can be given to justify such postulates absolutely. They come from ‘revelations’ in the minds of ‘geniuses,’ most likely with hints from Nature based on extensive careful observations. Their general validity can only be established through experimental verification. If numerous rigorously derived logical consequences of a very small set of postulates all agree with experimental observations without exception, one is inclined to accept these postulates as correct descriptions of the laws of Nature and use them confidently to explain and predict other natural phenomena. Quantum mechanics is no exception. It is based on a few postulates. For the purpose of the present discussion, we begin with three basic postulates involving: the “state functions,” “operators,” and “equations of motion.” In this chapter, this set of basic postulates and some of the corollaries and related definitions of terms are introduced and discussed. We will first simply state these postulates and introduce some of the related mathematical tools and concepts that are needed to arrive at their logical consequences later. To those who have not been exposed to the subject of quantum mechanics before, each of these postulates taken by itself may appear puzzling and meaningless at first. It should be borne in mind, however, that it is the collection of these postulates as a whole that forms the foundations of quantum mechanics.
Abstract In the previous six chapters, we focused on understanding the basic eigenstates of various Hamiltonians by solving the time independent Schrödinger equation. While the Hamiltonians seemed relatively simple, the solutions are frequently used. However, in practice, most of the important problems today do not make extensive use of the spatial solutions except when a specific number is needed. The focus on most problems today makes extensive use of operators where their behavior is governed by the new design rules that we must follow to do anything novel with quantum systems. The design rules are sometimes referred to as the postulates. Postulates are mathematical statements of specific rules. We will present five postulates in this chapter and then in later chapters augment them with three more. In the context of this math, we will continue to work with functions like ψ(r,t), but we will introduce Dirac notation such as ∣ψ(t)〉. The remaining chapters in this text work almost exclusively in Dirac notation and with operators.