The harmonic oscillator is, by some margin, the most important system in physics. This is partly because its easy and we can solve it. And partly because, under the right circumstances, pretty much anything else can be made to look like a bunch of coupled harmonic oscillators. In this chapter, we look at what happens when a bunch of harmonic oscillators – or springs – are connected to each other.

The essence of dimensional analysis is very simple: if you are asked how hot it is outside, the answer is never “2 o’clock”. You’ve got to make sure that the units, or “dimensions”, agree. In this chapter, we understand what it means for quantities to have dimensions and how getting to grips with this can help solve problems without doing any serious work.

Symmetries are a key idea in physics. In the classical world, they are associated to conservation laws, courtesy of Emmy Noether. The same, and more, is true in the quantum world. In this chapter we explore how symmetries manifest themselves in quantum mechanics. Special attention will be given to time evolution and the role of SU(2) and angular momentum

This chapter deals with advanced topics for a multivariate Langevin and Fokker–Planck dynamics. For systems with multiplicative noise it is shown that neither the drift term in the Langevin equation nor the discretization parameter can be determined uniquely. If one of the two is fixed, the other one is determined. In contrast, the Fokker–Planck equation, which contains the physically observable distribution is unique. Experimental data for a particle near a wall illustrate the relevance of space-dependent friction. Martingales are introduced for a Langevin dynamics with a nonlinear expression of entropy production as a prominent example that with Doob’s optimal stopping theorem leads to universal results of its statistics. Finally, underdamped Langevin dynamics is described by the Klein–Kramers equation, for which entropy production is determined by the irreversible currents. A multi-time-scale analysis recovers the Smoluchowski equation in the overdamped limit even in the presence of an inhomogeneous temperature for which an anomalous contribution to entropy production is found.

Whereas RRR has been successfully applied to a broad range of problems, the example in this chapter shows that our understanding of the algorithm is far from complete.

Our goal in this chapter is to look more closely at the underlying mathematical formalism of quantum mechanics. We will look at the quantum state, how it evolves in time, and what it means to interrogate the state by performing a measurement. It is here that we meet the famed Heisenberg uncertainty principle.

Optimal protocols transform a given initial distribution into a given final one in finite time with a minimal amount of work or entropy production. We first analyze this optimization paradigmatically for a driven harmonic oscillator for which analytical results can be obtained. For a general Langevin dynamics, it is shown that the optimal protocol can be realized through a time-dependent potential with no need to use a nonconservative force. In contrast for discrete systems, nonconservative driving decreases the thermodynamic costs. For a broader perspective, we introduce concepts from information geometry which deals with the statistical manifold of distributions. The Fisher information provides a metric on this manifold from which the distance between two distributions as the minimal length connecting them can be derived. Speed limits yield relations between these quantities referring to an initial and a final distribution and the entropy production associated with the transformation of the former into the latter. For slow processes, cost along the optimal protocol or path is bounded by the distance between these distributions and the inverse of the allocated time.

The bipartisan problem formulation is introduced, where relatively simple sets A and B are designed that hold solutions in their intersection.

This chapter deals with processes both from a macroscopic, thermodynamic point of view and from a dynamical perspective. For the latter, a class of processes is introduced that can be described through a Hamiltonian description with a time-dependent external control parameter. It is shown how the expressions of work and heat from classical thermodynamics can be obtained as an appropriate average over an initial distribution. The second law inequality relating work and free energy can then be proven as a consequence of a master inequality. With well-specified additional assumptions, second law inequalities for heat exchange and entropy production are derived.

Until now, we’ve only considered the motion of a single particle. If our goal is to understand everything in the universe, that’s a little limiting. In this section, we take a small step forwards: we will describe the dynamics of multiple interacting particles. Among other things, this will highlight the importance of the conservation of momentum and angular momentum.

Classical mechanics starts with Newtons three laws, among them the famous F=ma. But these laws are not quite as transparent as they may seem. In this chapter, we introduce the laws and provide some commentary. We will also learn about Galileos ideas of relativity, a precursor to the much more shocking ideas of Einstein that come later.

Laura M. Chalk (later, Laura Rowles, 1904−1996) was the first woman to complete a PhD in physics at McGill University in Montreal, Canada. Her doctoral research on the quantum phenomenon called the Stark effect, under the supervision of J. Stuart Foster, produced the earliest experimental test of Erwin Schrödinger’s wave mechanics. After a brief stint as a postdoctoral fellow at King’s College London, she chose to return home and dedicate herself to teaching and marriage. This paper aims to fully recover Chalk’s work and explore why the Foster−Chalk experiment was overlooked in physics historiography. It considers the Stark effect’s significance in quantum physics and the impact of gender on her personal trajectory. Shaped by personal choice, systemic discrimination, and acceptance of societal norms, Chalk Rowles’ story highlights the paradoxes faced by women in a culturally disembodied yet male-dominated field, and reflects broader themes of gender and identity in the history of women in physics.

Quantum particles, like happy families, are all the same. In fact, not only are they the same. They are literally indistinguishable. This has deep and important consequences that are fleshed out in this chapter.