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 $m=-j,-j+1,\dots ,j-1,j$.

There are many systems in nature that are made up of several particles of the same species. These particles all have the same mass, charge and spin, and need to be treated as identical particles. For instance, the electrons in an atom are identical particles. Identical particles cannot be distinguished by measuring their intrinsic properties. While this is also true for classical particles, the laws of classical mechanics allow us to follow the trajectory of each individual particle, i.e. their time evolution in space.

Very few problems in quantum mechanics can be solved exactly. For example, in the case of the helium atom, including the inter-electron electrostatic repulsion term in the Hamiltonian changes the problem into one which cannot be solved analytically. Perturbation theory provides a method for finding approximate energy eigenvalues and eigenstates for a system whose Hamiltonian is of the form.

In this chapter we are going to set up the formalism to describe observables in quantum mechanics. This is an essential part of the formulation of the theory, as it deals with the description of the outcome of experiments. Beyond any theoretical sophistication, a physical theory is first and foremost a description of natural phenomena; therefore it requires a very precise framework that allows the observer to relate the outcome of experiments to theoretical predictions. As we will see, this is particularly true for quantum phenomena. The necessary formalism is very different from the intuitive one used in classical mechanics. A subtle point is that the state of the system is not an observable by itself. As seen in the , the state of the system is specified by a complex vector.

There are various calculational methods beyond the perturbation theory of thethat can be applied in specific circumstances to give either exact or approximate results. In this chapter some of the most common methods are explained. We start with the Rayleigh–Ritz variational method that can be used to obtain an upper-bound estimate of the ground-state energy of a quantum-mechanical system. Next we examine multi-electron atoms. In such a case simple application of perturbation theory becomes difficult and more needs to be done.