What exactly is a spin

d. What are bosons and fermions?

In contrast to conventional macroscopic particles, quantum objects have no precisely specified location and no precisely specified speed. Rather, they are "smeared" over a certain location (typically the deBroglie wavelength) and speed range. The principle behind it is called Heisenberg's uncertainty principle. It was formulated in 1927 by Werner Heisenberg in the context of quantum mechanics. From this it follows, however, that if you bring two particles so close together that their wave functions touch, they in principle become indistinguishable. You can't even tell them apart by their position. So a many-particle wave function has to be found that does not change its properties when two or more individual particles are exchanged.

This fact led to the development of symmetric and anti-symmetric many-particle wave functions. These wave functions ensure that what is required above actually occurs; namely that an exchange of particles does not cause any physical change. At this point it should be clear that all particles in nature can basically be divided into two categories. Particles with a symmetric wave function are called bosons, while those with an anti-symmetric wave function are called fermions.

So far there is no physical theory that can predict which particles are bosons and which fermions. The spin statistics theorem only provides a theoretical justification for the empirical finding that particles with integer spin are bosons and those with half-integer spin are fermions. Spin is a property of quantum mechanical objects; in simplified terms, it can be imagined as the particle's own rotation, just as the earth rotates around its own axis. However, this view is not entirely correct. Spin is covered in more detail in a later section.

In some respects the two types of particles have opposite properties. The most important property that distinguishes the two from each other is the fact that two fermions can never assume the same quantum state. So two fermions, in one system, can never have the same physical properties.

Any atom can be used to make this clear. Atoms consist of a nucleus and an electron shell. The electrons in the shell have a spin ½ which makes them fermions. In the atom, the wave functions of the individual electrons in the shell overlap very strongly. This makes it necessary to describe it by an anti-symmetrical many-particle wave function. Hence, two electrons cannot be in the same state. With this knowledge one can also explain why the electrons in the atom form different orbitals around the nucleus. Viewed naively, one could assume that all electrons have to collect in the lowest orbital, since this has the lowest energy. Instead, in reality there is only room for two electrons with opposite spins in the lowest orbital.

In contrast, bosons tend to be more in oneCollect state. This is generally counteracted by thermal movements of the bosons, so that at a finite temperature not all particles of a boson gas assume the lowest energy state. At absolute zero, however, all bosons in the system should be there.

As an example of a boson, one can use the photon, which represents the quantum of the electromagnetic field, or, to put it simply, a light particle. A lot of photons are generated in a laser. The vast majority of them have the same frequency and move in the same direction. They thus assume the same quantum state. Only a small part of the emitted photons have a different frequency or direction of propagation.

The properties of bosons and fermions described above can be summarized in the distribution functions; they are dealt with in the next but one section. In order to understand the distribution functions, however, the free energy and the associated chemical potential must first be introduced. So the next section deals with the question:

What are the free energy and chemical potential?