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.

Chapter 3 starts out with a physics motivation, as well as a mathematical statement of the problem that will be tackled in later sections. After a brief discussion of analytical differentiation, the bulk of the chapter is devoted to increasingly better finite-difference approximations, like the forward difference and the central difference. These are explicitly derived using Taylor expansions, and also applied to second derivatives and to points on a grid. A section introduces the useful tool of Richardson extrapolation, which reappears in later chapters. The chapter also includes an original discussion of automatic differentiation, which is built up from the concept of dual numbers. The chapter is rounded out by a physics project, which studies the kinetic energy in single-particle quantum mechanics, and a problem set. The physics project involves different wave functions and provides the groundwork for the project in the integrals chapter.

Two extra-Galactic surveys are considered. The first takes observations of nearly 200 GMCs across a small sample of local galaxies in the CANON CO survey. In spite of the local nature of the sample, results confirm essential facts about molecular gas distribution in galaxies other than our own, including a confirmed linear relationship between GMC virial mass and CO luminosity, which implies a constant CO-H2 conversion factor and supports a virialization assumption. The second survey, PHANGS-ALMA (Physics at High Angular Resolution in Nearby Galaxies with the Atacama Large Millimetre-submillimetre Array), maps CO emission from galaxies up to 17 Mpc away, with resolutions of 1″–1.5″ encompassing active star-forming galaxies down to total stellar masses ~5 × 109 M☉. Within 11 of those target galaxies considered here, the results offer tens of thousands of measurements at GMCs scales between 20 and 130 pc, comparable to Galactic-scale observations, and one outcome is confirmation of a positive correlation between GMC surface densities and velocity distributions.

Whenever there is flow of water in an alluvial channel, the water entrains and transports sediment. The entrainment and transport of sediment require work that flow must perform, and to perform the work requires energy. The water has potential energy due to elevation, which is transformed to kinetic energy part of which is utilized to do this work and part is dissipated to overcome boundary and bed friction. The time rate of potential energy expenditure is the stream power, which plays a fundamental role in the evolution of a fluvial system. Using the theory of stream power, this chapter derives constraints for hydraulic geometry.

The work–energy theorem in a general reference frame is presented. It relates the change of the mechanical energy of the system in a finite time interval to the impulse work of the interaction forces (and inertia forces, if the frame is a non-Galilean one). Different expressions for the calculation of the kinetic energy of a rigid body are provided. The concept of conservative system and conservative force are introduced, and from them, that of potential energy. The particular case of gravitational energy and energy associated with linear springs is studied. A section is devoted to the study of equilibrium configurations, and their stability, of systems of one degree of freedom. Finally, the impossibility of perpetual motion is proved. The rotation stability of a free rigid body is analyzed through Poinsot’s ellipsoid in an appendix.

Should cars be fuelled by petrol or electricity? Should houses use electricity sourced from a renewable resource such as solar panels, or from a coal-fired power station? Energy is a significant topic from both a global and local perspective, with topical election issues such as these polarising the community. Energy cannot be seen directly – we can only experience its effects as it interacts with living things and materials, In the F–6 energy learnng progression in the Australian Curriculum, there is a focus on sound energy in Years F–2, on heat energy in Years 3–4 and on electrical energy in Years 5–6. Investigating the properties of a range of energy forms such as light, sound, heat, movement and electrical energy, and how they interact with materials, provides students with entry points to identify solutions to real-life issues. This chapter presents some common alternative conceptions associated withenergy, then describes three integrated STEM projects for primary school children.