The final chapter is a largely non-technical overview of economic and political aspects of wind energy policy. The cost of wind energy is assessed in terms of Levelised Cost of Energy (LCoE), with equations given in full and simplified forms. Using a large database, historic installed costs for UK wind both onshore and offshore are given, from the earliest projects to the present day. The observed trends are discussed. Operational and balancing costs are outlined, the latter reflecting the intermittency of wind power. LCoE estimates are made for a range of installed costs and output capacity factors at typical discount rates and compared with current generation prices. The chapter considers the economics of onsite generation with the example of a private business using wind energy to offset demand; the energy displacement and export statistics are extrapolated to compare with a national scenario for 100% renewable electricity generation. The topic of ownership is introduced and examined in the context of the UK’s first community-owned windfarm. The chapter concludes with a brief review of UK renewable energy policy, which originated with legislation to protect the nuclear power industry.

Chapter 7 considers structural loading and response of horizontal-axis machines, with some theoretical background and illustrative measurements from different wind turbine types. The chapter begins with a recap of the dynamics of a single degree of freedom system, leading into a discussion of multi-DOF systems and modal analysis. The cyclic loads affecting a wind turbine structure are described, including wind shear, tower shadow, and rotationally sampled turbulence. The concepts of stochastic and deterministic loading are explained and the principle of aerodynamic damping illustrated. Qualitative descriptions are given of gyroscopic, centrifugal, and electromechanical loading. The phenomenon of blade edgewise stall vibration is explained, with discussion of mechanical damper solutions. The last part of the chapter draws on an early experimental campaign in which the dynamic loading on a full-scale wind turbine was measured and compared with the results of software simulation. Results from the same trials also demonstrate the difference in rotor thrust loading arising from positive and negative pitch control. The chapter concludes with a brief summary of fatigue prediction methods.

If you’ve worked through Chapters 1 and 2, you’ve already seen several references to the Schrödinger equation and its solutions. As you’ll learn in this chapter, the Schrödinger equation describes how a quantum state evolves over time, and understanding the physical meaning of the terms of this powerful equation will prepare you to understand the behavior of quantum wavefunctions. So this chapter is all about the Schrödinger equation, and you can read about the solutions to the Schrödinger equation in Chapters 4 and 5.

In the first section of this chapter, you’ll see a “derivation” of several forms of the Schrödinger equation, and you’ll learn why the word “derivation” is in quotes. Then, in Section 3.2, you’ll find a description of the meaning of each term in the Schrödinger equation as well as an explanation of exactly what the Schrödinger equation tells you about the behavior of quantum wavefunctions. The subject of Section 3.3 is a time-independent version of the Schrödinger equation that you’re sure to encounter if you read more advanced quantum books or take a course in quantum mechanics.

To help you focus on the physics of the situation without getting too bogged down in mathematical notation, the Schrödinger equation discussed in most of this chapter is a function of only one spatial variable (x). As you’ll see in later chapters, even this one-dimensional treatment will let you solve several interesting problems in quantum mechanics, but for certain situations you’re going to need the three-dimensional version of the Schrödinger equation. So that's the subject of the final section of this chapter (Section 3.4).

Origin of the Schrödinger Equation

If you look at the introduction of the Schrödinger equation in popular quantum texts, you’ll find that there are several ways to “derive” the Schrödinger equation. But as the authors of those texts invariably point out, none of those methods are rigorous derivations from first principles (hence the quotation marks). As the brilliant and always-entertaining physicist Richard Feynman said, “It's not possible to derive it from anything you know. It came out of the mind of Schrödinger.”

So if Erwin Schrödinger didn't arrive at this equation from first principles, how exactly did he get there? The answer is that although his approach evolved over several papers, from the start Schrödinger clearly recognized the need for a wave equation from the work of French physicist Louis de Broglie.

The conclusions reached in the previous chapter concerning quantum wavefunctions and their general behavior are based on the form of the Schrödinger equation, which relates the changes in a particle or system's wavefunction over space and time to the energy of that particle or system. Those conclusions tell you a great deal about how matter and energy behave at the quantum level, but if you want to make specific predictions about the outcome of measurements of observables such as position, momentum, and energy, you need to know the exact form of the potential energy in the region of interest. In this chapter, you’ll see how to apply the concepts and mathematical formalism described in earlier chapters to quantum systems with three specific potentials: the infinite rectangular well, the finite rectangular well, and the harmonic oscillator.

Of course, you can find a great deal more information about each of these topics in comprehensive quantum texts and online. So the purpose of this chapter is not to provide one more telling of the same story; instead, these example potentials are meant to show why techniques such as taking the inner product between functions, finding eigenfunctions and eigenvalues of an operator, and using the Fourier transform between position and momentum space are so important in solving problems in quantum mechanics. As in previous chapters, the focus will be on the relationship between the mathematics of the solutions to the Schrödinger equation and the physical meaning of those solutions. And although we live in a universe with (at least) three spatial dimensions in which the potential energy may vary over time as well as space, most of the essential physics of quantum potential wells can be understood by examining the one-dimensional case with time-independent potential energy. So in this chapter, the Schrödinger equation is written with position represented by x and potential energy by V(x).

Infinite Rectangular Potential Well

The infinite rectangular well is a potential configuration in which a quantum particle is confined to a specified region of space (called the “potential well”) by infinitely strong forces at the edges of that region. Within the well, no force acts on the particle. Of course, this configuration is not physically realizable, since infinite forces do not occur in nature. But as you’ll see in this section, the infinite rectangular potential well has several features that make this a highly instructive configuration.

This book has one purpose: to help you understand the Schrödinger equation and its solutions. Like my other Student's Guides, this book contains explanations written in plain language and supported by a variety of freely available online materials. Those materials include complete solutions to every problem in the text, in-depth discussions of supplemental topics, and a series of video podcasts in which I explain the most important concepts, equations, graphs, and mathematical techniques of every chapter.

This Student's Guide is intended to serve as a supplement to the many comprehensive texts dealing with the Schrödinger equation and quantum mechanics. That means that it's designed to provide the conceptual and mathematical foundation on which your understanding of quantum mechanics will be built. So if you’re enrolled in a course in quantum mechanics, or you’re studying modern physics on your own, and you’re not clear on the relationship between wave functions and vectors, or you want to know the physical meaning of the inner product, or you’re wondering exactly what eigenfunctions are and why they’re so important, then this may be the book for you.

I’ve made this book as modular as possible to allow you to get right to the material in which you’re interested. Chapters 1 and 2 provide an overview of the mathematical foundation on which the Schrödinger equation and the science of quantum mechanics is built. That includes generalized vector spaces, orthogonal functions, operators, eigenfunctions, and the Dirac notation of bras, kets, and inner products. That's quite a load of mathematics to work through, so in each section of those two chapters you’ll find a “Main Ideas” statement that concisely summarizes the most important concepts and techniques of that section, as well as a “Relevance to Quantum Mechanics” paragraph that explains how that bit of mathematics relates to the physics of quantum mechanics.

So I recommend that you take a look at the “Main Ideas” statements in each section of Chapters 1 and 2, and if your understanding of those topics is solid, you can skip past that material and move right into a term-byterm dissection of the Schrödinger equation in both time-dependent and timeindependent form in Chapter 3. And if you’re confident in your understanding of the meaning of the Schrödinger equation, you can dive into Chapter 4, in which you’ll find a discussion of the quantum wavefunctions that are solutions to that equation.

Chapter 6 considers wind turbine control, including supervisory control, power limiting, starting and stopping, electrical power quality, and sector management. The importance of accurate yaw control is discussed in terms of energy capture and cyclic loading, and an active yaw system is illustrated. The main focus of the chapter is real-time power control, and the chapter builds on the aerodynamic and electrical concepts covered previously in Chapters 3–5. The differences between stall and pitch regulation are explained, in the latter case in the context of both constant and variable-speed operation. Power measurements from constant-speed and variable-speed pitch controlled machines illustrate the superior accuracy of the latter. Control block diagrams are given for both methods, with qualitative explanation of the principles. The procedure for starting and stopping different wind turbine types is explained, and the advantages of pitch control in this context are illustrated. The chapter includes a short description of sector management, a control strategy based on external factors such as wind speed and direction, and used for noise reduction, shadow flicker prevention, or fatigue mitigation.

Chapter 5 deals with electrical issues and is broadly divided in two. The first half explains the operating principles of the several different types of generator found on wind turbines and their influence on dynamics and electrical power quality. Generator types are illustrated schematically and their characteristics explained using simple physical principles. Geared and gearless (direct drive) generators are discussed, and there is a brief historical review of generator developments. The second half of the chapter deals with electrical networks and further examines the issue of power quality. The importance of reactive power and how modern generators can manipulate it to aid voltage stability are explained; the role of external devices such as Statcoms, SVCs, and pre-insertion resistors is also discussed in this context. Measurements from a MW-scale wind turbine illustrate voltage control via reactive power management over a period of several days. The challenge of low grid strength is illustrated with a practical example of a small wind farm development on a rural network with low fault level. The chapter concludes with a brief discussion of wind turbine lightning protection.

This chapter contains a broad overview of the technical and environmental issues to be addressed in the construction of onshore wind energy projects. The former include ecological considerations, including birds and mammals; the requirements of typical pre-construction ornithological surveys are described with an example. Public safety and acceptance is discussed in the context of catastrophic damage to wind turbines, visual impact, shadow flicker, and noise nuisance. In the last case, equations and simple rules for noise assessment are given in the context of typical planning guidelines. Sound power levels for a range of commercial wind turbines are compared, and empirical relationships are given relating noise to rated output, rotor size, and tip speed. Risks to aviation are discussed, covering aircraft collision and interference to radar systems, including both primary and secondary surveillance radars. The concept of ‘stealthy’ wind turbine blades is discussed and described in outline. Other siting criteria include avoidance of RF and microwave communications beams and television interference. Rules are given to avoid interference, while minimising required separation distances.

If you’re wondering how the abstract vector spaces, orthogonal functions, operators, and eigenvalues discussed in Chapters 1 and 2 relate to the wavefunction solutions to the Schrödinger equation developed in Chapter 3, you should find this chapter helpful. One reason that relationship may not be obvious is that quantum mechanics was developed along two parallel paths, which have come to be called the “matrix mechanics” of Werner Heisenberg and the “wave mechanics” of Erwin Schrödinger. And although those two approaches are known to yield equivalent results, each offers benefits in elucidating certain aspects of quantum theory. That's why Chapters 1 and 2 focused on matrix algebra and Dirac notation while Chapter 3 dealt with plane waves and differential operators.

To help you understand the connections between matrix mechanics and wave mechanics, the first section of this chapter explains the meaning of the solutions to the Schrödinger equation using the Born rule, which is the basis for the Copenhagen interpretation of quantum mechanics. In Section 4.2, you’ll find a discussion of quantum states, wavefunctions, and operators, along with an explanation of several dangerous misconceptions that are commonly held by students attempting to apply quantum theory to practical problems.

The requirements and general characteristics of quantum wavefunctions are discussed in Section 4.3, after which you can see how Fourier theory applies to quantum wavefunctions in Section 4.4. The final section of this chapter presents and explains the form of the position and momentum operators in both position and momentum space.

The Born Rule and Copenhagen Interpretation

When Schrödinger published his equation in early 1926, no one (including Schrödinger himself) knew with certainty what the wavefunction ψ represented. Schrödinger thought that the wavefunction of a charged particle might be related to the spatial distribution of electric charge density, suggesting a literal interpretation of the wavefunction as a real disturbance – a “matter wave.” Others speculated that the wavefunction might represent some type of “guiding wave” that accompanies every physical particle and controls certain aspects of its behavior. Each of these ideas has some merit, but the question of what is actually “waving” in the quantum wavefunction solutions to the Schrödinger equation was very much open to debate.