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.

The topic of this chapter is the wave function – what it is, how it is to be interpreted and how information can be extracted from it. To this end, the notion of operators in quantum physics is introduced. And the statistical interpretation called the Born interpretation is discussed. This discussion also involves terms such as expectation values and standard deviations. The first part, however, is dedicated to a brief outline of how quantum theory came about – who were the key people involved, and how the theory grew out of a need for understanding certain natural phenomena. Parallels are drawn to the historical development of our understanding of light. At a time when it was generally understood that light is to be explained in terms of travelling waves, an additional understanding of light consisting of small quanta turned out to be required. It was in this context that Louis de Broglie introduced the idea that matter, which finally was known to consist of particles – atoms – must be perceived as waves as well. Finally, formal aspects such as Dirac notation and inner products are briefly addressed. And units are introduced which allow for convenient implementations in the following chapters.

All examples seen in the preceding chapters have dealt with a single particle. In this chapter, the theory is expanded to systems with several identical particles. Here, ‘many’ in practice means two. However, this does allow the introduction of several central aspects. Perhaps the most important one is spin, which is the topic of the first part. Central elements in this context are the Stern–Gerlach experiment and the Pauli matrices. The characteristics of these matrices are studied in some detail as they play crucial roles in the remainder of the book. The concept of entanglement in quantum physics is introduced – exemplified using both the two-particle spin wave function and the combined spin–space wave function for a single particle. Due to the Pauli principle, the importance of spin and exchange symmetry in a many-body context is hard to underestimate. The fact that identical particles are indistinguishable has implications for the symmetry of the wave function. This, in turn, has significant consequences for the structure of the system – including its ground state. This is investigated by performing calculations of energy estimates. Most of these apply the variational principle, but also the notion of self-consistent field and the Hartree–Fock method are introduced.

Simulating a quantum system exposed to some explicitly time-dependent influence differs from that of quantum systems without time dependence in the Hamiltonian. In the latter case, one can, as in Chapter 2, study the full time evolution by means of a relatively simple time-evolution operator, whereas small time steps must be imposed to study the more dynamic case in which also the Hamiltonian changes in time. The first examples of such address the comparatively simple cases of one and two spin-½ particles exposed to magnetic fields. In this context, the rotating wave approximation is introduced. Later, the spatial wave function of a one-dimensional model of an atom exposed to a laser pulse is simulated. To this end, so-called Magnus propagators are used. It is also outlined how the same problem may be recast as an ordinary differential equation by expanding the wave function in the so-called spectral basis consisting of the eigenstates of the time-independent part of the Hamiltonian. The time evolution in this context may be found by more standard methods for ordinary differential equations. Also, the two-particle case if briefly addressed before what is called the adiabatic theorem is introduced. Its validity is checked by implementing a specific, dynamical system.

This chapter aims to illustrate how quantum theory provides useful technological solutions – applications that may be more integrated in our everyday lives than we tend to think. Some applications lend themselves to a particularly straightforward outline through examples already seen in the preceding chapters. These include scanning tunnelling microscopy and emission spectroscopy, which utilize tunnelling and energy quantization, respectively. Prior knowledge and readymade implementations allow these applications to be studied in a quantitative manner. Also, nuclear magnetic resonance is, albeit in a somewhat simplified model, studied quantitatively – within the framework of an oscillating spin-½ particle developed in Chapter 5. The remainder of the chapter is dedicated to quantum information technology. Also in this context, the notion of one or two spin-½ particles is applied frequently. A spin-½ particle is one possible realization of a quantum bit, and it serves well as a model even in cases when quantum bits are implemented differently. After having introduced some basic notions, two specific protocols for quantum communication are studied in some detail. The last part of the chapter addresses adiabatic quantum computing. This technology is studied in a manner that lies close to the last example of Chapter 5.