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.

This project focuses on the Initial Value Problem (IVP) for ordinary differential equations with the application of multipoint recursion schemes. The effectiveness and convergence of these schemes are explored and subsequently applied to examine the properties of a compound pendulum, specifically the dependence of the oscillation period on energy. The chapter then focuses on Newton’s laws of motion, laying the foundation for understanding the motion equation. The project uses a simple pendulum to illustrate the concept, looking at how changes in amplitude affect the period of harmonic oscillations. Numerical methods, such as recursive methods based on local extrapolation, are then employed to derive formulas. The project concludes by discussing the integration of Runge–Kutta methods and implicit schemes to solve the equations. This project ultimately questions the viability of the pendulum as a standard unit of time, adding value to ongoing discussions in physics and mathematics education.

This project explores the phenomenon of coupled harmonic oscillators, which have a broad relevance in fields such as mechanics, atomic and quantum physics, optics, electronics, and biology. The paper expands on a model of a harmonic oscillator, using numerical methods for solving differential equations to analyse systems of coupled oscillators. The study focuses on two illustrative cases: a one-dimensional system of two point masses suspended from springs, and a system of two simple pendulums moving in the same plane. The derived mathematical equations of motion provide a comprehensive framework for understanding the behaviour of such systems. Through computational experimentation, the project aims to elucidate the oscillatory behaviour of these systems depending on different parameters and their coupling, particularly focusing on energy transfer between the oscillators. The Runge–Kutta algorithm is employed for solving the initial value problem (IVP) for the ordinary differential equations (ODE) governing these systems. The project underscores the versatility of the harmonic oscillator model by showing that different physical systems can be described by the same mathematical model.

This project involves the application of molecular dynamics (MD) to a simple two-dimensional planetary system consisting of two planets and a fixed star. The primary focus is to construct a MD code using Newton’s law of universal gravitation as the interaction law and the Verlet algorithm for solving the initial value problem. The project examines the gravitational interaction described by Newton’s laws, focusing on the law of universal gravitation and its application to the planetary system. It further explores the principle of equivalence, the concept of conservative force, and the effective potential energy of the system. The discussion also covers the reduction of a single planet motion to one dimension, which offers insights into the trajectory of the planetary system. Finally, the project outlines the numerical approach using the Verlet algorithm for simulating the motion of the planets. The comprehensive understanding of the gravitational interactions and the computational techniques provide a solid foundation for the study of complex dynamical systems.

This chapter discusses the application of the variational principle and finite element (FE) methods to electrostatic systems, using a cylindrical capacitor as a representative example. The variational principle, which identifies the solution that minimises a system’s total energy, provides a foundation for numerical techniques such as the FE methods. These methods offer flexibility in the selection of points in the independent variable space, which can be adapted according to the expected function behaviour. The system of focus is a capacitor comprising two coaxial metallic cylinders, chosen for its high symmetry and real-world relevance in areas like telecommunication cables. The chapter presents the Poisson’s equation, demonstrates the variational principle, and introduces the FE method. A numerical example of a cylindrical capacitor is also provided, illustrating how to convert the problem into one of functional minimisation using Poisson’s equation and then solve it using the FE method. The Gauss–Seidel iterative minimisation procedure is used in the solution process. The discussion provides a foundation for extending these methods to 2D and 3D systems in subsequent chapters.