In this chapter we describe a technique to deal with identical particles that is called second quantization. Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields, fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level.

Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.

In the same series of experiments which led to the discovery of superconductivity, Kamerlingh Onnes was the first who succeeded in cooling helium below its boiling point of 4.2 K. In 1908, he was cooling this liquefied helium further in an attempt to produce solid helium. But since helium only solidifies at very low temperatures (below 1 K) and high pressures (above 25 bar), Kamerlingh Onnes did not succeed, and he turned his attention to other experiments.

It was, however, soon recognized that at 2.2 K (a temperature reached by Kamerlingh Onnes), the liquid helium underwent a transition into a new phase, which was called helium II. After this observation, it took almost 30 years to discover that this new phase is actually characterized by a complete absence of viscosity (a discovery made by Kapitza and Allen in 1937). The phase was therefore called superfluid. In a superfluid, there can be persistent frictionless flows – superflows – very much like supercurrents in superconductors. This helium II is, up to now, the only superfluid which is available for experiments.

The most abundant isotope of helium, 4He, is a spinless boson. Given this fact, the origin of superfluidity was almost immediately attributed to Bose condensation, which was a known phenomenon by that time.

Preface

Courses on advanced quantum mechanics have a long tradition. The tradition is in fact so long that the word “advanced” in this context does not usually mean “new” or “up-todate.” The basic concepts of quantum mechanics were developed in the twenties of the last century, initially to explain experiments in atomic physics. This was then followed by a fast and great advance in the thirties and forties, when a quantum theory for large numbers of identical particles was developed. This advance ultimately led to the modern concepts of elementary particles and quantum fields that concern the underlying structure of our Universe. At a less fundamental and more practical level, it has also laid the basis for our present understanding of solid state and condensed matter physics and, at a later stage, for artificially made quantum systems. The basics of this leap forward of quantum theory are what is usually covered by a course on advanced quantum mechanics.

Most courses and textbooks are designed for a fundamentally oriented education: building on basic quantum theory, they provide an introduction for students who wish to learn the advanced quantum theory of elementary particles and quantum fields. In order to do this in a “right” way, there is usually a strong emphasis on technicalities related to relativity and on the underlying mathematics of the theory. Less frequently, a course serves as a brief introduction to advanced topics in advanced solid state or condensed matter.

Most introductory courses on physics mention the basics of atomic physics: we learn about the ground state and excited states, and that the excited states have a finite life-time decaying eventually to the ground state while emitting radiation. You might have hoped for a better understanding of these decay processes when you started studying quantum mechanics. If this was the case, you must have been disappointed. Courses on basic quantum mechanics in fact assume that all quantum states have an infinitely long life-time. You learn how to employ the Schrödinger equation to determine the wave function of a particle in various setups, for instance in an infinitely deep well, in a parabolic potential, or in the spherically symmetric potential set up by the nucleus of a hydrogen atom. These solutions to the Schrödinger equation are explained as being stationary, the only time-dependence of the wave function being a periodic phase factor due to the energy of the state. Transitions between different quantum states are traditionally not discussed and left for courses on advanced quantum mechanics. If you have been disappointed by this, then this chapter will hopefully provide some new insight: we learn how matter and radiation interact, and, in particular, how excited atomic states decay to the ground state.

Our main tool to calculate emission and absorption rates for radiative processes is Fermi’s golden rule (see Section 1.6.1). We thus start by refreshing the golden rule and specifying its proper use. Before we turn to explicit calculations of transition rates, we use the first part of the chapter for some general considerations concerning emission and absorption, those being independent of the details of the interaction between matter and electromagnetic radiation. In this context we briefly look at master equations that govern the probabilities to be in certain states and we show how to use those to understand properties of the equilibrium state of interacting matter and radiation. We show that it is even possible to understand so-called black-body radiation without specifying the exact interaction Hamiltonian of matter and radiation.

In this chapter (summarized in Table 4.1) we present a simple model to describe the phenomenon of magnetism in metals. Magnetism is explained by spin ordering: the spins and thus magnetic moments of particles in the material tend to line up. This means that one of the spin projections is favored over another, which would be natural if an external magnetic field is applied. Without the field, both spin projections have the same Zeeman energy, and we would expect the same numbers of particles with opposite spin, and thus a zero total magnetic moment. Magnetism must thus have a slightly more complicated origin than just an energy difference between spin states.

As we have already seen in Chapter 1, interaction between the spins of particles can lead to an energy splitting between parallel and anti-parallel spin states. A reasonable guess is thus that the interactions between spin-carrying particles are responsible for magnetism. The conduction electrons in a metal can move relatively freely and thus collide and interact with each other frequently. Let us thus suppose that magnetism originates from the interaction between these conduction electrons.

We first introduce a model describing the conduction electrons as non-interacting free particles. Within this framework we then propose a simple wave function which can describe a magnetic state in the metal. We then include the Coulomb interaction between the electrons in the model, and investigate what this does with the energy of our trial wave function.

This chapter presents a short introduction to relativistic quantum mechanics that conforms to the style and methods of this book. In this context, relativity is a special kind of symmetry. It turns out that it is a fundamental symmetry of our world, and any understanding of elementary particles and fields should be necessarily relativistic – even if the particles do not move with velocities close to the speed of light. The symmetry constraints imposed by relativity are so overwhelming that one is able to construct, or to put it better, guess the correct theories just by using the criteria of beauty and simplicity. This method is certainly of value although not that frequently used nowadays. Examples of the method are worth seeing, and we provide some.

We start this chapter with a brief overview of the basics of relativity, refresh your knowledge of Lorentz transformations, and present the framework of Minkowski spacetime. We apply these concepts first to relativistic classical mechanics and see how Lorentz covariance allows us to see correspondences between quantities that are absent in non-relativistic mechanics, making the basic equations way more elegant. The Schrödinger equation is not Lorentz covariant, and the search for its relativistic form leads us to the Dirac equation for the electron. We find its solutions for the case of free electrons, plane waves, and understand how it predicted the existence of positrons. This naturally brings us to the second quantization of the Dirac equation. We combine the Dirac and Maxwell equations and derive a Hamiltonian model of quantum electrodynamics that encompasses the interaction between electrons and photons. The scale of the interaction is given by a small dimensionless parameter which we have encountered before: the fine structure constant α ≈ 1/137 ≪ 1. One thus expects perturbation theory to work well. However, the perturbation series suffer from ultraviolet divergences of fundamental significance. To show the ways to handle those, we provide a short introduction to renormalization. The chapter is summarized in Table 13.1.

In this chapter we explain the procedure for quantizing classical fields, based on the preparatory work we did in the previous chapter. We have seen that classical fields can generally be treated as an (infinite) set of oscillator modes. In fact, for all fields we considered in the previous chapter, we were able to rewrite the Hamiltonian in a way such that it looked identical to the Hamiltonian for a generic (mechanical) oscillator, or a set of such oscillators. We therefore understand that, if we can find an unambiguous way to quantize the mechanical oscillator, we get the quantization of all classical fields for free!

We thus start considering the mechanical harmonic oscillator and refresh the quantum mechanics of this system. The new idea is to associate the oscillator with a level for bosonic particles representing the quantized excitations of the oscillator. We then turn to a concrete example and show how the idea can be used to quantize the deformation field of the elastic string discussed in Section 7.2. In this case, the bosonic particles describing the excitations of the field are called phonons: propagating wave-like deformations of the string. As a next example, we revisit the low-lying spinfull excitations in a ferromagnet, already described in Section 4.5.3. Here, we start from a classical field theory for the magnetization of the magnet, and show how the quantization procedure leads to the same bosonic magnons we found in Chapter 4. Finally, we implement the quantization technique for the electromagnetic field, leading us to the concept of photons – the quanta of the electromagnetic field.