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.

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.

Advances in technologies that were unimaginable a century ago have helped in establishing the current high standards of living. Undoubtedly, the oil and gas industry has played a pivotal role in this respect. Thanks to the advent of the petroleum industry, the use of oil and gas has created new factories and revolutionized industries such as transportation and power generation for more than a century. Liquid fuels have impacted transportation and have made various communities closer. The reliance on liquid and gaseous fuels has affected the lives of every person in the world with the invention of air transportation and personal vehicles.

In the 1990s, British writers began using “transparency” as a portmanteau word to describe that desirable state of organizational management and governance characterized by candor, openness, honesty, clarity, legal compliance, and full disclosure (Handy, 1990). At first, the word didn’t take hold on this side of the Atlantic, perhaps because it was too vague and philosophical for American tastes in managerial buzz words (which tend to run more to the precise and practical).

Given the rapid rate of technological innovation and a desire to be proactive in addressing potential ethical challenges that arise in contexts of innovation, engineers must learn to engage in value-sensitive design – design that is responsive to a broad range of values that are implicated in the research, development, and application of technologies. One widely-used tool is Life Cycle Assessment (LCA). Physical products, as with organisms, have a life cycle, starting with extraction of raw materials, and including refining, transport, manufacturing, use, and finally end-of-life treatment and disposal. LCA is a quantitative modeling framework that can estimate emissions that occur throughout a product’s life cycle, as well as any harmful effects that these emissions have on the environment and/or public health. Importantly, LCA tools allow engineers to evaluate multiple types of environmental and health impacts simultaneously and are not limited to a single endpoint or score. However, LCA is only useful to the extent that its models accurately include the full range of values implicated in the use of a technology, and to the extent that stakeholders, from designers to decisionmakers, understand and are able to communicate these values and how they are assigned. Effective LCA requires good ethical training to understand these values.

There was a time when we were all six-sigma-ing. We did so because Jack Welch had bought into the six-sigma phenomenon and he had created a phenomenally performing General Electric (GE). Then we moved along from good to great to the search for excellence to becoming great by choice to whatever superlative Jim Collins told us was the way to a company that was built to last. Then someone moved our cheese. We had no time for that because we were just one-minute managers. We smoothed earnings, incentivized employees, and created three tiers of employees – including getting rid of the bottom tier of employees, whether they deserved termination or kudos. We all wanted to be part of the Fortune 100, the Fortune Most Admired Companies, even as we were led by Fortune CEOs and CFOs of the year – many of whom ended up doing time.

Modern engineering and technology have allowed us to connect with each other and even to reach the moon. But technology has also polluted vast areas of the planet and empowered surveillance and authoritarian governments with dangerous tools. There are numerous cases where engineers and other stakeholders routinely ask what they are capable of inventing, and what they actually should invent. Nuclear weapons and biotechnology are two examples. But when analyzing the transformations arising from less controversial modern socio-technological tools – like the Internet, smartphones, and connected devices, which augment and define our work and social practices – two very distinct areas of responsibility become apparent. On the one hand, a question arises around the values and practices of the engineers who create the technologies. What values should guide their endeavors and how can society promote good conduct? On the other hand, there are questions regarding the effects of people using these technologies. While engineering and design choices can either promote or hinder commendable social behavior and appropriate use, this chapter will focus on the first question.

As technology becomes more powerful, intelligent, and autonomous, its usage also creates unintended consequences and ethical challenges for a vast array of stakeholders. The ethical implications of technology on society, for example, range from job losses (such as potential loss of truck driver jobs due to automation) to lying and deception about a product that may occur within a technology firm or on user-generated content platforms. The challenges around ethical technology design are so multifaceted that there is an essential need for each stakeholder to accept responsibility. Even policymakers who are charged with providing the appropriate regulatory framework and legislation about technologies have an obligation to learn about the pros and cons of proposed options.

There is a delicate balance associated with ethics and privacy in “enterprise continuous monitoring systems.” On the one hand, it can be critical to enterprises to continuously monitor the ethical behavior of different agents, and thus, facilitate enterprise risk management, as noted in the KPMG quote. In particular, continuous monitoring systems help firms monitor related internal and external agents to make sure that the agents hired by or engaged by the enterprise are behaving ethically. However, on the other hand, such continuous monitoring systems can pose ethical and privacy risks to those being monitored and provide risks and costs to the company doing the monitoring. For example, inappropriate information can be assembled, stored, and inferred about a range of individuals. Thus, information obtained by continuous monitoring generally should follow privacy principles that require that the data be up-to-date and conform to the purpose for which the data was originally gathered, and other constraints.

When the US Army established the Institute for Creative Technologies (ICT) at the University of Southern California in 1999, the vision was to push the boundaries of immersive technologies for the purpose of enhancing training and education; not only for the military but also for the rest of society. Over the past two decades great progress has been made on the technologies that support this vision. Breakthroughs in graphics, computer vision, artificial intelligence (AI), affective computing, and mixed reality have already transformed how we interact with one another, with digital media, and with the world. Yet this is in many ways only a starting point, since the application of these technologies is just beginning to be realized. The potential for making a positive impact on individuals and society is great, but there is also the possibility of misuse. This chapter describes some of the capabilities underlying the emerging field of immersive digital media; provides a couple of examples of how they can be used in a positive way; and then discusses the inherent dangers from an ethical standpoint.

Each year hundreds of new biomedical devices and therapies are developed to attempt to solve unmet medical needs.  However, many fail due to unforeseen challenges of complex ethical, regulatory, and societal issues.  We propose that a number of these issues can be effectively transformed into drivers of innovation for medical solutions if ethical analysis is considered early, iteratively, and comprehensively in the research and development process.

This chapter is based on ethical distinctions. Creating clarity in ethical thought depends on the clarity of distinctions we make in discussing ethical issues. Achieving clarity and consistency in ethical behavior requires understanding some basic distinctions.

In an essay about his science fiction, Isaac Asimov reflected that “it became very common … to picture robots as dangerous devices that invariably destroyed their creators.” He rejected this view and formulated the “laws of robotics,” aimed at ensuring the safety and benevolence of robotic systems. Asimov’s stories about the relationship between people and robots were only a few years old when the phrase “artificial intelligence” (AI) was used for the first time in a 1955 proposal for a study on using computers to “solve kinds of problems now reserved for humans.” Over the half-century since that study, AI has matured into sub-disciplines that have yielded a constellation of methods that enable perception, learning, reasoning, and natural language understanding.

The development and popularity of computer-mediated communication (CMC), social network sites (SNSs), and social media communication (SMC) sparked twenty-first-century ethical dilemmas (Patching & Hirst, 2014; Barnes, 2003). At the heart of social media ethical concerns are data privacy and ownership. The fallout from the Cambridge Analytica data breach on Facebook, which followed a class action settlement in 2012 over the Beacon program, offers clear evidence that lack of user consent over gathering and disseminating information is a long-standing problem (Terelli, Jr. & Splichal, 2018). Facebook appears to have made the problem worse by allowing user data access to outside, third-party program applications (“apps”), and granting user friends the ability to further weaken privacy (Stratton, 2014).

Twenty-first-century innovations in the technical fields designed for human consumption and ultimately as daily life necessities such as personal robots, intelligent implants, driverless cars, and drones require innovations in ethical standards, laws and, rules of ethics. Ethical issues around robots and artificial (AI) intelligence, for example, present a new set of challenges about the new capabilities they afford. These capabilities outpace law and policy in ethics. Tesla and Space X CEO Elon Musk recently warned the governors of the United States that “robots will do everything better than us” and that “AI is a fundamental existential risk for human civilization.” He called for the proactive government regulation of AI, “I think by the time we are reactive in AI regulation, it’s too late” (Domonoske, 2017)