This chapter begins a five-chapter mathematical introduction to quantum field theory, appropriate for upper-level undergraduate science or engineering students, or those with some mathematical training who would like to know what the “real” theory of quantum mechanics is. In this chapter, the basics of Dirac notation are presented. The last part of this chapter shows how the uncertainty principle of quantum mechanics is derived.

This chapter surveys several different mathematical methods for time-dependent change of quantum states using quantum field theory. The Bloch sphere method is introduced, which can be used to show the physics discussed in Chapter 3, that electronic transitions, or “jumps,” are not instantaneous.

This chapter begins a short, two-chapter section on calculations that specifically impact the philosophy of quantum mechanics. A quantitative discussion of the famous Einstein–Podalsky–Rosen (EPR) experiment is given, as well as a mathematical discussion of problems with the many-worlds hypothesis, the Bohmian pilot-wave hypothesis, and the “transactional” hypothesis for interpreting quantum mechanics.

This chapter starts out a short, two-chapter section on very basic mathematics of quantum mechanics, appropriate for those who have taken undergraduate science or engineering courses. The method of “unit analysis” is used as a way of getting at when quantum mechanics will play a role in the behavior of things.

Given the philosophical problems of the Copenhagen interpretation, several other approaches to interpreting quantum mechanics have been proposed over the years. This chapter surveys four of these approaches, namely the many-worlds hypothesis, Bohmian “pilot waves,” positivist approaches, and spontaneous collapse of the quantum wave function. Problems with each of these approaches are discussed.

After having shown in previous chapters that wave-particle duality is not a fundamental problem for quantum mechanics, this chapter introduces the really strange effect of quantum mechanics, namely “nonlocal correlations” that appear to act over long distances faster than the speed of light. The “Copenhagen” interpretation of quantum mechanics is introduced, which puts human knowledge in a special role, and some of the philosophical objections to it.

This chapter explains what we mean by “fields” and “waves” in physics, and argues that quantum waves are just as “real” as other waves we experience in daily life, such as water waves and sound waves.

This chapter addresses generalizations of the Schrödinger equation. It tries to convey that the Schrödinger equation is not the whole story when it comes to quantum physics. This is illustrated by expanding the framework in two rather orthogonal directions: relativistic quantum physics and open quantum systems. The former is introduced by taking the Klein–Gordon equation as the starting point, before shifting attention to the Dirac equation. Its time-independent version is solved numerically for a one-dimensional example, and its relation to the Schrödinger equation is derived. Also here, the Pauli matrices play crucial roles. The notion of open quantum systems is motivated by the fact that it is hard to keep a quantum system completely isolated from its surroundings – and that this necessitates a different approach than the one provided by wave functions. To this end, reduced density matrices and the notion of master equations are introduced. It is explained why master equations of the form of the generic Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) equation are desirable. Two particular phenomena following this equation are studied quantitatively: amplitude damping for a single quantum bit system and particle capture in a confining potential. Again, these examples draw directly on previous ones.

The closing chapter aims to sum up some of the experiences, albeit in a rather overarching way. It is emphasized that, while the book spans rather widely, much of what has been presented is a bit like scratching the surface. Still, the tools developed should form a good basis for further work within quantum sciences. And, hopefully, the book has worked as a way of getting to know a bit of the quantum nature of the micro cosmos. In the preceding chapters, questions related to quantum foundations have, to a large extent, been evaded. Addressing the measurement problem and alternative interpretations attempts to mitigate this. A few topics are listed which are essential to quantum physics but are not properly addressed in this book. This includes quantum field theory, perturbation theory, density functional theory and quantum statistics. Finally, there are provide suggestions for further reading.

In this chapter, the aim is to visualize wave dynamics in one dimension as dictated by the Schrödinger equation. The necessary numerical tools are introduced in the first part of the chapter. Via discretization, the wave function is represented as a column vector and the Hamiltonian, which enters into the Schrödinger equation, as a square matrix. It is also seen how different approximations behave as the numerical wave function reaches the numerical boundary – where artefacts appear. This numerical framework is first used to see how a Gaussian wave packet would change its width in time and, eventually, spread out. Two waves interfering is also simulated. And wave packets are sent towards barriers to see how they bounce back or, possibly, tunnel through to the other side. In the last part of the chapter, it is explained how quantum measurements provide eigenvalues as answers – for any observable physical quantity. This, in turn, is related to what is called the collapse of the wave function. It is also discussed how a quantity whose operator commutes with the Hamiltonian is conserved in time. Finally, the concept of stationary solutions is introduced in order to motivate the following chapter.

This chapter starts out by introducing the energy eigenvalue equation – the time-independent Schrödinger equation. Firstly, the notion of energy quantization is introduced by semi-analytical means. A particle is confined within a rectangular well. It is seen, eventually, that the corresponding solutions to the time-independent Schrödinger equation can only exist for a few, specific energies. This phenomenon is also seen for other physical quantities, such as angular momentum and charge. Next, the situation in which the potential experienced by a quantum particle is periodic is given particular attention. It may be studied numerically by minor adjustments of the framework already developed. The case of a periodic potential is important as, to a large extent, it forms the basis for understanding solid state physics. Since the Hamiltonian is Hermitian, its eigenvectors form an orthonormal set in which any state or wave function may be expanded. This is exploited in the last part of the chapter, which is dedicated to determining ground states – the energy eigenstate of minimal energy. This is done in two ways – by using what is called the variational principle and by so-called propagation in imaginary time. These methods are implemented for several examples and compared to full solutions.

In Chapter 7 an effective non-Hermitian Hamiltonian was introduced. This topic is elaborated upon in this chapter. Examples are studied which use the artificial but useful notion of non-Hermitian – for both dynamical and stationary cases. One is a revisit of the example from Chapter 6 on a model atom exposed to a laser pulse. A complex absorbing potential is introduced to enable calculation on a numerical domain smaller than the actual physical system. The same technique is also applied to the examples seen in Chapter 2 on a wave packet hitting a barrier. By introducing a double barrier, the notion of resonances emerges. In this example, resonances are manifested in pronounced peaks in the transmission probability. If the same system is described by combining the time-independent Schrödinger equation with outgoing boundary conditions, the same peaks may be identified by complex energies. Discussion follows of the interpretation of the imaginary part as the width and the lifetime of a resonance. Finally, another type of resonances is studied, namely doubly excited states, and their relation to the physical phenomena of the Auger–Meitner effect and that of capture via dielectronic recombination. This is done in a rather non-technical way.