O NCE WE leave the realm of either one dimensional or single particle systems, visualizing quantum wavefunctions becomes more challenging. Suppose, for example, we choose to model a single particle confined to a two-dimensional surface? This is analogous to the 1D systems we encountered in Chapter 3 except that the particle can now move in more than one dimension. The wavefunction then has to depend on both x and y . The trouble is that when we attempt to visualize such a wavefunction we can no longer represent the complex value as a 2D phasor, since after using the x and y direction for the position in space, we only have the z direction left for visualization! Since a phase has two independent parts, either (real, imaginary) or (magnitude, phase). We need some way to visualize these two components distributed over some area. One approach is to use color to represent phase, and height in the z direction to represent magnitude. Another would be to place small objects at a distribution of spatial points and use the properties (e.g., color, size, orientation) to represent various aspects of the wavefunction. In this chapter we'll explore a combination of these approaches and develop some insights from the results.
O NCE WE leave the realm of either one dimensional or single particle systems, visualizing quantum wavefunctions becomes more challenging. Suppose, for example, we choose to model a single particle confined to a two-dimensional surface? This is analogous to the 1D systems we encountered in Chapter 3 except that the particle can now move in more than one dimension. The wavefunction then has to depend on both x and y . The trouble is that when we attempt to visualize such a wavefunction we can no longer represent the complex value as a 2D phasor, since after using the x and y direction for the position in space, we only have the z direction left for visualization! Since a phase has two independent parts, either (real, imaginary) or (magnitude, phase). We need some way to visualize these two components distributed over some area. One approach is to use color to represent phase, and height in the z direction to represent magnitude. Another would be to place small objects at a distribution of spatial points and use the properties (e.g., color, size, orientation) to represent various aspects of the wavefunction. In this chapter we'll explore a combination of these approaches and develop some insights from the results.
O NE OF the most basic ideas of quantum mechanics is that the condition of a system is represented mathematically by a wavefunction . This is a complex valued mathematical function of that tells us everything we can know about the system. In order to make predictions we apply mathematical operations to the wavefunction that tell us the probability of various outcomes that we can compare to experimental results. The wavefunction is a complex valued function. In other words when you evaluate it for a particular set of inputs (e.g., position, time, momentum, energy, etc.) the output is a complex number. The magnitude of the complex number gives information about the relative likelihood of a particular outcome. That seems pretty odd I'm sure. The best way to get comfortable with the notion is to play around with it. Let's consider an example.
A quantum system's state is described by a complex wave function, which can be supported on different topologies - for example, one-dimensional open space, a closed one-dimensional ring, or a two-dimensional surface of a sphere. Interactive tools represent the quantum state of a single particle constrained to different topologies, while stressing the underlying superposition principle, which is independent of topology. The insight gained by visualizing the same phenomena in different geometrical contexts contributes to the students' ability to abstract, which is the key to understanding quantum phenomena in more complex systems.
We have developed an app for iOS-based smart-phones/tablets that allows a 3-D, complex phase-based colorful visualization of hydrogen atom wave functions. Several important features of the quantum behavior of atomic orbitals can easily be made evident, thus making this app a useful companion in introductory modern physics classes. There are many reasons why quantum mechanical systems and phenomena are difficult both to teach and deeply understand. They are described by equations that are generally hard to visualize, and they often oppose the so-called “common sense” based on the human perception of the world, which is built on mental images such as locality and causality. Moreover students cannot have direct experience of those systems and solutions, and generally do not even have the possibility to refer to pictures, videos, or experiments to fill this gap. Teachers often encounter quite serious troubles in finding out a sensible way to speak about the wonders of quantum physics at the high school level, where complex formalisms are not accessible at all. One should however consider that this is quite a common issue in physics and, more generally, in science education. There are plenty of natural phenomena whose models (not only at microscopic and atomic levels) are of difficult, if not impossible, visualization. Just think of certain kinds of waves, fields of forces, velocities, energy, angular momentum, and so on. One should also notice that physical reality is not the same as the images we make of it. Pictures (formal, abstract ones, as well as artists' views) are a convenient bridge between these two aspects.
To a casual ostrich the world of quantum physics in one dimension may sound a little one-dimensional, suitable perhaps for those with an unhealthy obsession for the esoteric. Nothing of course could be further from the truth. The field is remarkably rich and broad, and for more than fifty years has thrown up innumerable challenges. Theorists, realising that the role of interactions in 1D is special and that well known paradigms of higher dimensions (Fermi liquid theory for example) no longer apply, took up the challenge of developing new concepts and techniques to understand the undoubted pecularities of one-dimensional systems. And experimentalists have succeeded in turning pipe dreams into reality, producing an impressive and ever increasing array of experimental realizations of 1D systems, from the molecular to the mesoscopic---spin and ladder compounds, organic superconductors, carbon nanotubes, quantum wires, Josephson junction arrays and so on. Many books on the theory of one-dimensional systems are however written by experts for experts, and tend as such to leave the non-specialist a touch bewildered. This is understandable on both fronts, for the underlying theoretical techniques are unquestionably sophisticated and not usually part of standard courses in many-body theory. A brave author it is then who aims to produce a well rounded, if necessarily partial, overview of quantum physics in one dimension, accessible to a beginner yet taking them to the edge of current research, and providing en route a thorough grounding in the fundamental ideas, basic methods and essential phenomenology of the field. It is of course the brave who succeed in this world, and Thierry Giamarchi does just that with this excellent book, written by an expert for the uninitiated. Aimed in particular at graduate students in theoretical condensed matter physics, and assumimg little theoretical background on the part of the reader (well just a little), Giamarchi writes in a refreshingly relaxed style with infectious enthusiasm for his subject, and readily combines formal instruction with physical insight. The result is a serious, pedagogical yet comprehensive guide to the fascinating and important field of one-dimensional quantum systems, for which many a graduate student (and not a few oldies) will be grateful. The first half of the book, chapters 1--5, is devoted to a coherent presentation of the essential concepts and theoretical methods of the field. After a basic introduction to the unique behaviour of interacting electrons in one dimension, and to early fermionic approaches to the problem, Giamarchi turns to the technique of bosonization, introducing chapter 3 with a Marxist quote: `A child of five would understand this. Send for a child of five.' This most powerful technique is presented in a step by step fashion, and serious perusal of the chapter will benefit all ages since bosonization is used extensively throughout the rest of the book. The same is true of chapter 3 where a phenomenological and physically insightful introduction is given to the Luttinger liquid---the key concept in the low-energy physics of one-dimensional systems, analogous to the Fermi liquid in higher dimensions. Chapter 4 deals with what the author calls `refinements', or complications of the sort theorists in particular welcome; such as how the Luttinger liquid description is modified by the presence of long-ranged interactions, the Mott transition (`we forgot the lattice Gromit'), and the effects of breaking spin rotational invariance on application of a magnetic field. Finally chapter 5 describes various microscopic methods for one dimension, including a brief discussion of numerical techniques but focussing primarily on the Bethe ansatz---the famous one-dimensional technique others seek to emulate but whose well known complexity necessitates a relatively brief discussion, confined in practice to the spin-1/2 Heisenberg model. In the second half of the book, chapters 6--11, a range of different physical realizations of one-dimensional quantum physics are discussed. According to taste and interest, these chapters can be read in essentially any order. Spin systems are considered in chapter 6, beginning with spin chains---Jordan--Wigner, the bosonization solution---before moving to frustration, the spin-Peierls transition, and spin ladders; and including experimental examples of both spin chain and ladder materials. Chapters 7 and 8 deal with interacting lattice fermions, the former with single chain problems, notably the Hubbard, t-J and related models; and the latter with coupled fermionic chains, from finite to infinite, including a fulsome discussion of Bechgaard salts (organic conductors) as exemplars of Luttinger liquid behaviour. The effect of disorder in fermionic systems is taken up in chapter 9, and here the reader may react: interacting systems are tough enough, why make life harder? But disorder is always present to some degree in real systems---quantum wires, for example, discussed briefly in the chapter---and its effects particularly acute in one dimension. It simply cannot be avoided, even if the problem of interacting, disordered one-dimensional systems is still a long way off being solved. The penultimate chapter deals with the topical issues of boundaries, isolated impurities and constrictions, with a primary focus on mesoscopic examples of Luttinger liquids, notably carbon nanotubes and edge states in the quantum Hall effect. Finally `significant other' examples of Luttinger liquids, namely interacting one-dimensional bosons, are considered in chapter 11; which concludes with a discussion of bosonization techniques in the context of quantum impurities in Fermi liquids---the x-ray, Kondo and multichannel Kondo problems. The quality of the product attests to the fact that writing this impressive tome was a labour of love for the author. Anyone with a serious interest in getting to grips with one-dimensional quantum systems simply needs the book on their shelves---and will have great fun reading it too.
In the previous chapter we saw that for a particle in a one-dimensional potential well, physically acceptable solutions to the one-dimensional Schro¨dinger equation are possible only for particular discrete values of the total energy. Moreover, in the case of an electron in a well of atomic dimensions, the spacings between these energy levels are in qualitative agreement with the separations experimentally observed in atoms; we also interpreted the square of the wave function as a probability distribution for the position of the particle. These ideas led us to phenomena, such as quantum-mechanical tunnelling, that have been observed experimentally. The real world, however, is three-dimensional and, although one-dimensional examples often provide useful insights and analogies, we shall have to extend our theory into three dimensions before we can make quantitative predictions of most experimental results. In the present chapter, therefore, we shall set up the three-dimensional Schro¨dinger equation and obtain its solutions in a number of cases, culminating in a discussion of the hydrogen atom where we shall find that theory and experiment agree to a remarkable degree of accuracy.
In (1) the Wigner transform of an oscillator is calculated and the high n energy level limit taken using the WKB approximation yielding a result proportional to delta(p - p(x)) + delta(p+p(x)) where p(x) = sqrt(2m(E-.5kxx)). The Wigner transform involves quantum wavefunctions which for bound systems include +/- p components which interfere i.e. a(p)(exp(ipx) + exp(-ipx) for a(p)=a(-p). Classically, a bound particle moves to the right during one time interval and to the left during the other so there is no interference of +/-p. The quantum wavefunction retains +/-p interference for all energy levels and so does not portray the full physical information of the bound system i.e. leads to the two delta functions. For high n (energy level), crests and troughs are very narrow compared to the length of the system, approximating dx units of length. Thus, uncertainty in time is only within these tiny crests/troughs, otherwise there is a distinct time position relationship as in classical physics, but the wavefunction W(x) does not show it. How then may the wavefunction be interpreted? We argue the time independent wavefunction, say for a bound state, represents an averaged steady stream picture which is often not the physical picture. For example, a bound particle is not a steady stream average nor is a finite number of photons hitting an interface of media with different indices of refraction/reflection. Nevertheless, the steady stream approach with time removed may be used to solve the problem i.e. ultimately find a one dimensional spatial density (which averages out the original two dimensional probability of exp(ipx), energy levels etc. For instance, the energy levels of a bound system do not depend on following the particle in time. We raise this issue because we think it may have implications on the calculation of entropy in bound single particle quantum systems. In general, a spatial Shannon’s entropy is defined as: - W(x)W(x) ln[W(x)W(x)] (one dimensional problem). W(x) (wavefunction) is real because p/-p pieces interfere in a steady stream picture. A single bound particle, however, is not a steady stream picture. exp(ipx) is a dynamic probability equal to cos(px)+i sin(px) to indicate the direction of motion. Simply because exp(ipx)+exp(-px) removes the complex portion (i.e. for a steady state picture with a(p)=a(-p)) does not mean that this represents the full physical picture especially as energy levels become high and crests/troughs become narrower with respect to the overall system length it seems there is no justification for the steady stream interference picture.. In other words, using a real W(x) in a Shannon’s entropy expression treats one dimensional probability W(x) = bound wavefunction, but this is not the full probability in the picture, we argue. Full probability is two dimensional and demonstrates direction for many energy levels (especially high ones). For a free particle exp(ipx), cos(px) and sin(px) are essentially the same distribution. Thus cos(px)cos(px) describes the probability distribution in x. For W1(x)=Sum p>0 a(p)exp(ipx), Real(W1(x)) is not equal to Imaginary(W1(x)) because one does not have a simple shift of the real distribution to obtain the imaginary one. The real distribution is chosen to represent probability because the interference of exp(ipx), exp(-ipx) terms removes the imaginary piece, but it is removed only on average. If physically exp(ipx) and exp(-ipx) should not interfere, then one may need to take some kind of average of Real(W1(x))Real(W1(x)) and Imaginary(W1(x))Imaginary(W1(x)) and use this in Shannon’s entropy.