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.
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 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 .
Classical mechanics and electromagnetismThe mechanics of motion formulated by Newton and contemporaries in the 17th century is the foundation of contemporary physics.It is immensely successful, explaining and quantifying a vast array of phenomena, ranging from planetary motion to the flow of fluids.By the latter part of the 19th century the study of electric and magnetic fields and their inclusion in the framework of Newtonian physics appeared near completion.Since motion and electromagnetism are central to the failures of Newtonian physics in the atomic realm, it is useful to review the essential ideas, concepts, and the implicit assumptions of what is now referred to as classical physics.Many of the results presented in this chapter are assumed to be familiar to the reader, but some concepts are more thoroughly presented because of their relevance to quantum mechanics. Newtonian mechanicsFor simplicity initial consideration is limited to a single point particle of inertial mass m.A point particle is treated as infinitesimally small and indivisible.It is possible to formulate Newtonian mechanics in terms of continuous matter distributions, referred to as continuum mechanics, but these will not be considered in this text.In that regard, point particles are often used as an approximation to the constituents of real matter in Newtonian mechanics.A point particle's position therefore coincides with a point in space at every given moment of time, parameterized with t, and can be described mathematically by a vector x from the choice of origin to that point.As the point particle moves its position is given by the time-dependent trajectory x t ( ).An important assumption of Newtonian mechanics is that all observers are equipped with identical meter sticks and synchronized clocks, and they all agree upon the current position at the time t, although the choice of each observer's origin is arbitrary.The Cartesian components of the vector trajectory are denoted x t ( ) i , where the subscript i runs over the dimensions of the space under consideration.In one dimension the subscript is
Chapter 1 was concerned with the need for a quantal description of microscopic phenomena. The thrust of Chapter 3 is an analysis and comparison of a classical and a quantum system; some mathematical generalizations of the analysis are presented as well. Chapter 4 provides an introduction to Hilbert-space concepts that are the mathematical framework of quantum theory, while, in Chapter 5, the axioms of quantum mechanics for pure states are stated and illustrated. The present chapter is intended to be a bridge between those chapters and Chapter 1. Here we review many of the concepts of classical mechanics that have relevance in quantum mechanics. Equally important, if not more so, we stress the lack of equivalence between a number of classical concepts and their quantal counterparts. This inequivalence is emphasized because of the vocabulary common to classical and quantum mechanics. Classical physics is, of course, the source for much of the vocabulary in quantum physics. However, the ways in which the words are used in the two domains often differ substantially, while the concepts are almost necessarily not identical. Some classical concepts have no direct quantal counterparts. For example, velocity and force are not primary quantal entities. This fact means that the quantal definitions of linear and angular momentum cannot be the same as those in classical mechanics. Similarly, the concept of a “state” in classical physics – the collection of the generalized positions and generalized velocities or momenta – does not carry over to quantum physics.
The first chapter described measurements involving the spin of an electron. We saw that if you measure the spin in the vertical direction, you don't obtain a continuum of values, but just two of them: Either the electron has its north pole vertically upward, or it is vertically downward. If we measure the spin first in the vertical direction and then once more in the same direction, we obtain exactly the same result for both measurements. If the first measurement shows that the electron has its north pole upward, then so will the second measurement. We also saw that if we measure first in the vertical direction and then in the horizontal direction, the electrons will have spin N and S in direction 90° each with probability of one half. It doesn't matter what the first measurement is; the second measurement will give a random choice of either N or S. The second chapter introduced the mathematics of linear algebra. The goal of this chapter is to combine these previous two chapters, giving a mathematical model that describes the measurement of spin. We will then show how this relates to qubits. But before we start this description, we introduce the mathematics of probability.
In this second chapter we will review the basic mathematical and physical facts about quantum mechanics and establish physical units and notation. Those readers already familiar with the subject can safely jump to the next chapter. An attempt has been made to make the presentation in this chapter as elementary as possible, and yet present the basic facts that will be needed later. There are many beautiful and important topics which will not be touched upon such as self-adjointness of Schrödinger operators, the general mathematical structure of quantum mechanics and the like. These topics are well described in other works, e.g.. Much of the following can be done in a Euclidean space of arbitrary dimension, but in this chapter the dimension of the Euclidean space is taken to be three-which is the physical case-unless otherwise stated. We do this to avoid confusion and, occasionally, complications that arise in the computation of mathematical constants. The interested reader can easily generalize what is done here to the R d , d > 3 case. Likewise, in the next chapters we mostly consider N particles, with spatial coordinates in R 3 , so that the total spatial dimension is 3 N . A Brief Review of the Connection Between Classical and Quantum Mechanics Considering the range of validity of quantum mechanics, it is not surprising that its formulation is more complicated and abstract than classical mechanics. Nevertheless, classical mechanics is a basic ingredient for quantum mechanics. One still talks about position, momentum and energy which are notions from Newtonian mechanics.
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.