Symmetries play a central role in the study of physical systems. They dictate the choice of the dynamical variables used to characterise the system, lead to conservation laws and set constraints on the evolution of the system. We shall see explicit examples of these features in this chapter, and set up the mathematical framework to be able to have a unified formalism for describing generic symmetries.

Inwe have introduced the basic concepts of quantum mechanics, and studied them for some simple, yet relevant, one-dimensional systems. In this chapter we take another step towards the description of real physical phenomena and generalise the concepts introduced so far to systems that evolve in more than one spatial dimension. The generalisation is straightforward and it will give us the opportunity to review some of the key ideas about physical states, observables and time evolution. In the process, we will encounter and highlight new features that were not present for one-dimensional systems. Once again let us emphasise that three-dimensional in this context refers to the dimension of the physical space in which the system is defined, and not to the dimensionality of the Hilbert space of states; the latter clearly will depend on the type of system that we consider. The three-dimensional formulation will allow us to discuss more realistic examples of physical systems. It will be clear as we progress through this chapter that everything we discuss can be generalised to an arbitrary number of dimensions. In some physical applications, where a quantum system is confined to a plane, a two-dimensional formulation will be useful. More generally, it is instructive to think about problems in arbitrary numbers of dimensions. In this respect, it is fundamental to be able to work with vectors, tensors, indices, and all that. Problems and examples in this chapter should help develop some confidence in using an index notation to deal with linear algebra.

We have finally developed all the tools that are necessary to study the hydrogen atom (H atom) from a quantum-mechanical perspective. In this chapter we present a non-relativistic formulation of the problem, where the interaction is modelled with a static Coulomb potential.

One-dimensional systems are an ideal playground to test the concepts introduced so far in this book. Solving a few examples explicitly and discussing their physical interpretation will allow us to get familiar with the description of quantum systems and their dynamics. In this chapter we will solve Schrödinger’s equation for some simple one-dimensional potentials. Having found the solution of the mathematical equations, we will focus on how to extract physical information from these solutions. For these problems it is convenient to work in the position representation and describe the state of the system by its wave function.

One of the most common methods for studying the structure of matter and fundamental particles is to scatter particles and measure the outcome of the collision. For the collision between two particles, the problem is posed such that initially both particles are widely separated and moving towards each other but are assumed to evolve independently of any influence from each other. Then the two particles come within close proximity and interact with each other, resulting in their initial state being altered. Subsequently, the two particles in their altered form separate and some distance away this altered state is measured.

We now examine problems where the Hamiltonian depends on time. For such cases, energy is not conserved and so there are no stationary states. In general, for such problems it is very difficult to find exact solutions to the Schrödinger equation. Where progress can be made is in situations where time dependence is only in a small part of the Hamiltonian. Time-dependent perturbation theory is the formal approach to address such problems.

Having described the states and the observables of a quantum system, we shall now introduce the rules that determine their time evolution.

Now that we have developed the tools to describe three-dimensional systems, we are ready to introduce into our quantum-mechanical framework the concept of angular momentum. Recall that in classical mechanics angular momentum is defined as the vector product of position and momentum.

For observables like position and momentum, in quantum mechanics the quantum states in general do not give them an absolute existence. Their value in a particular system is generally only known once the measurement is made. Nevertheless, certain correlations can be present in a system. For a system that is made up of two or more parts that can be measured separately, such as at distinctly different spatial positions, the measurement of one part of the system may immediately imply what the measurement at another part of the system will be. This is a feature that can emerge in a quantum system which is entangled.

Quantum mechanics emerged as a natural extension of classical mechanics. As physics probed into the microscopic realm, it could be argued it would be almost impossible not to discover quantum mechanics. The spectra of atoms, the blackbody spectra, the photoelectric effect and the behaviour of particles through an array of slits had characteristically non-classical features. These phenomena were waiting their time for a theory to explain them. That does not diminish from the huge scientific insights of the founders of the subject. In physics, the great accomplishments come more often than not from insight rather than foresight. Knowing what will be the right physics 50 years into the future is a game of speculation. Recognising what is the important physics in the present and being able to explain it is the work of scientific insight. Thus, whereas we might say Democritus had great foresight millennia ago to envision the discrete nature of particles, it was Albert Einstein, Max Planck, Niels Bohr, Erwin Schrödinger, Werner Heisenberg, Paul Dirac, Max Born and Wolfgang Pauli who had the insights to develop quantum mechanics. And since their foundational work, our understanding of the physical world grew dramatically like never before.

Building on what we have discussed in the previous two chapters, we now turn to the problem of dealing with the addition of two angular momenta. For example, we might wish to consider an electron which has both an intrinsic spin and some orbital angular momentum, as in a real hydrogen atom. Or we might have a system of two electrons and wish to know what possible values the total spin of the system can take.

Quantum mechanics describes the behaviour of matter and light at the atomic scale, where physical systems behave very differently from what we experience in everyday life – the laws of physics of the quantum world are different from the ones we have learned in classical mechanics. Despite this ‘unusual’ behaviour, the principles of scientific inquiry remain unchanged: the only way we can access natural phenomena is through experiment; therefore our task in these first chapters is to develop the tools that allow us to compute predictions for the outcome of experiments starting from the postulates of the theory. The new theory can then be tested by comparing theoretical predictions to experimental results. Even in the quantum world, computing and testing remain the workhorses of physics.

Using the commutation relations for the components of the angular momentum, we have found that the allowed eigenvalues for ${\widehat{J}}^2$ are ${ħ}^{2}j(j+1)$, where $j=0,\frac{1}{2},1\frac{3}{2},\dots$. For each value of 𝑗, the eigenvalues of $J_z$ are $ħm$, with $m=-j,-j+1,\dots ,j-1,j$.