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.

Chapter 9, on siting and installation, considers some of the key steps leading to the successful installation of a wind energy project, whether a single machine or large array. A section on resource assessment considers site wind measurements, the IEC Wind Classification system, and the measure-correlate-predict (MCP) procedure for establishing long-term characteristics at a prospective site. Array interactions are described in terms of energy loss and increased turbulence: empirical models are given for predicting both effects, and wake influence is illustrated with field measurements from large and small arrays. The civil engineering aspects of project construction are examined, with description of different foundation types; simple rules are given for conventional gravity base design, with illustrations. The construction and environmental advantages of rock anchor foundations are described, and some examples are given. Transport, access, and crane operations are discussed. The use of winch erection is illustrated with the example of a 50kW machine. The chapter concludes with a short summary of the necessary electrical infrastructure between a wind turbine and the external grid network.

Chapter 4 extends the aerodynamic discussions of the preceding chapter to show how the rotor net loads (power, thrust, and torque) are developed. The dimensionless power coefficient (Cp) curve is introduced, and the relationship between rotor tip speed ratio and optimum solidity is explained. The variation of thrust loading with wind speed on an ideal pitch-controlled rotor is explained from simple theory and illustrated with measurements from a full-scale wind turbine. Equations governing the chord and twist distributions for an optimised blade are given and discussed in the context of some historic blade types, with illustrations. Rotor aerodynamic control is explained with reference to fixed-pitch stall regulation and variable blade pitch in both positive and negative senses. The influence of blade number is examined, with discussion of the advantages and disadvantages of one-, two-, and three-bladed wind turbines. The chapter concludes with a brief overview of alternative aerodynamic control devices, including tip vanes and ailerons, and the downwind rotor configuration (with examples).

This chapter is an overview of wind power meterorology at a relatively simple level without too much mathematical complexity. The origins of the wind are explained in the action of solar thermal radiation on the atmosphere, and the equation is given for the geostrophic wind at the top of the earth’s boundary layer. The role of the boundary layer in creating wind shear and turbulence near the earth’s surface is explained, and appropriate engineering equations are given to allow wind speed and turbulence to be estimated. Surface roughness and its relationship to turbulence and shear are explained. Experimental measurements are used to illustrate shear and turbulence for a range of different terrain types. The time and space dependency of wind speeds is also illustrated with site measurements, showing the long-term dependability of annual wind speeds, through the more variable monthly averages, to short-term turbulent variation. Gust factor is explained and illustrated as a function of turbulence intensity. The chapter includes high-resolution wind measurements taken during a storm in the Scottish Outer Hebrides, illustrating the extreme levels of turbulence arising in complex terrain.

The introductory chapter is a brief recap of the history and origins of wind power, from windmills in ancient times to today’s multi-megawatt turbines. Energy security has arguably been the historic driver for wind power, and it was a primary source of mechanical power until the advent of the Industrial Revolution, when it was superceded by coal and oil. The first electricity-generating wind turbines were built in the late nineteenth centry, and the technology was pursued most vigorously in Denmark, a country with limited energy reserves: the role of this country in creating the modern wind turbine is described. The worldwide energy crisis of the 1970s brought wind power into the frame internationally, and the pivotal role of legislation under President Carter in expanding the market for wind energy in the US and elsewhere is outlined. Since then, the rationale for wind power has expanded to include climate change, and the technology has grown exponentially in terms of global installation of wind power and the physical size of wind turbines. The chapter concludes by introducing some of the technological steps that have enabled this process, which are detailed in subsequent chapters.

The focus of this chapter is the trade-off between margin and volume. The analysis is couched within the context of monopoly price-setting. It is shown how to relate the profit-maximzing price to the elasticity demand. It also defines consumer surplus and shows how it may be calculated.The usefulness of these concepts is illustrated via application to the question of regulating a monopolist and double marginalization. The chapter ends by connecting cost functions to production functions.

This chapter introduces the basic models used to study imperfect competition: Bertrand, Cournot, Stackleberg, and Hotelling. Applications of these models are also described. Integrated into the exposition is an introduction to game theory and the concept of Nash equilibrium.