Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.

General derivation of the eigenvalues and eigenstates of the square and z-component of the angular momentum; relationship between the angular momentum and the harmonic oscillator in two dimensions; transformation of states and operators under rotations; algebraic derivation of the hydrogen atom spectrum.

Bound states in one-dimensional finite and infinite wells and delta-function potentials, and combinations of these, are obtained; the WKB method for bound states is introduced; the consequences of a parity-invariant potential for the eigenfunctions are derived.

Exchange and permutation operators; symmetrizer and antisymmetrizer; two bosons or two fermions in a central potential; scattering amplitudes of two identical particles in a central potential; the Fermi gas; theory of white dwarf stars; the Thomas-Fermi approximation for many-electron atoms.

Wave-particle duality and the Davisson-Germer experiment are briefly discussed; wave packets are defined and their features, including phase velocity, group velocity, and spreading, are examined; the stationary phase method is presented; free-particle wave functions are introduced, and the equivalence between coordinate- and momentum-space representations of these wave functions is emphasized.

Euler angles and rotation matrices; construction and properties of the rotation matrices; transformation of irreducible tensor operators under rotations; fine-structure of the hydrogen atom; hydrogen atom in a magnetic field: Zeeman and Paschen-Back effects; hyperfine structure of the hydrogen atom; tensor operators; time reversal and irreducible tensor operators.

Explicit solution of the hydrogen-like atom and isotropic harmonic oscillator radial equations by the technique of power series expansion; WKB derivation of the hydrogen-like spectrum; virial theorem; the two-dimensional isotropic harmonic oscillator in plane polar coordinates; the two-body problem and the center-of-mass and relative position and momentum operators.

Construction ofunitary operators inducing space and time translations, and rotations; the anti-unitary operator inducing time reversal; consequences of invariance under a symmetry transformation; periodic potentials and Bloch waves; the Kronig-Penney model; the ammonia molecule and broken parity symmetry; consequences of time reversal invariance on the scattering amplitude of spinless particles; Kramers degeneracy.

The Schroedinger equation for a particle in a potential is introduced and the general properties of its solutions are discussed; the uncertainity relations are derived; the Gram--Schimdt procedure for orthonormalizing a set of independent wave functions is introduced; the time evolution of the expectation values of the position and momentum operatorsfor a particle in a potential and in an electromagnetic field are derived.

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.