This chapter shows how particles arise naturally as an effect of waves, known as “resonance,” and that the particle concept, properly understood, is not somehow incompatible with the existence of waves. The definitions of “fermion” and “boson” fields, often associated with “matter” and “energy” particles, are introduced. The solidness of objects in our experience is a direct consequence of fermion wave properties.

This chapter introduces the formal “second quantization” method for bosons in quantum field theory. It is shown that phonons (sound particles) and photons (light particles) are simple extensions of the physics of a spring-like oscillator. The connection of boson states to classical waves is shown in a discussion of “coherent states.”

For a proper quantum mechanical description of multiple-particle systems, we must account for the indistinguishability of fundamental particles. The symmetrization postulate requires that the quantum state vector of a system of identical particles be either symmetric or antisymmetric with respect to exchange of any pair of identical particles within the system. Nature dictates that integer spin particles – bosons – have symmetric states, while half-integer spin particles – fermions – have antisymmetric states. The best-known manifestation of this is the Pauli exclusion principle, which limits the number of electrons in given atomic levels and leads to the structure of the periodic table.

There are no restrictions on how many bosons can occupy a single particle state, which has important consequences for their thermodynamic behaviour.Photons, quanta of the electromagnetic field, can be viewed as bosons with zero chemical potential, which allows the derivation of the thermodynamic properties of blackbody radiation, including the Stefan--Boltzmann Law.Non-interacting bosons with non-zero chemical potential can exhibit Bose--Einstein condensation at low temperatures, and interacting bosons may form a superfluid state.Low energy excitations in materials -- lattice vibrations (phonons) and spin waves (magnons) -- also behave as bosons, and are important for understanding the specific heat of materials at low temperatures.Of particular note is the Debye model which gives a simple account of the contributions of phonons to specific heat.

Unlike classical particles, quantum particles are indistinguishable.Fermions and bosons differ in their quantum statistics, and the consequences of this for their statistical mechanics are explored in the grand canonical ensemble.The Fermi--Dirac and Bose--Einstein distribution functions are derived, and utilized to write thermal averages using the density of states.