d. What are Bosons and Fermions?

Quantum objects in contrast to conventional macroscopic objects don’t have a specific location and velocity; instead they are smeared out over a certain region, typically the deBroglie wavelength and have a certain velocity distribution. The principle behind it is called Heisenberg uncertainty principle established by Werner Heisenberg. But this means if we bring particles so close together that their waves start to touch each other, they are principally indistinguishable. We can’t even distinguish between them due to their position. So if we make an operation with a quantum gas, let’s say rise the temperature the result should not depend on the indexing of the particles. Consequently the result of this operation should stay the same when we exchange the position of some of these particles.

This fact led to the invention of symmetrical and anti symmetrical wave functions. These wave functions assure the above demanded; that a particle exchange doesn’t change the result of an operation.Particles with a symmetric wave function are called Bosons; those with an anti symmetric wave function are called Fermions.

Till now there is no conclusive theoretical concept that predicts which particles are Bosons and which particles are Fermions, but empirically it seems that it has a lot to do with the spin of the particles. The spin is a property (inner degree of freedom) of quantum mechanical particles; one can imagine it as a rotation of the particle around its own axis, like the earth rotates around its axis, although this view is not correct at all. There are particles with fractional spin 1/2; 3/2; 5/2;…etc and with integer spin 1,2,3,4,…etc. It comes out that particles with integer spin have a symmetric wave function and are called Bosons and that such with fractional spin have anti symmetric wave functions and are called Fermions. The Spin-statistics theroem gives a theoretical justification for this observation, although it cannot be treated as a proof as it needs a lot of assumtions which are not proven by themselves.

In some aspects Bosons and Fermions have opposite features. The most important aspect is that two Fermions can never occupy the same quantum state.

As an example we can take any atom from the periodic table. Atoms consist of a nucleus and a electron shell.  Electrons have spin ½ and are therefore Fermions. Due to their fermionic nature they cannot occupy the same quantum state, that’s why they build up different orbits around the atom, otherwise it would be hard to explain why all the electrons in an atom do not collect in the lowest orbital as it has the lowest energy, which is always favored in nature.

In contrast Bosons love to occupy the same quantum state. This is generally avoided due to thermal excitation of a the Bose gas at finite temperatures. However at 0K all Bosons in a gas should occupy the lowest energy state.

As an example we can take Photons which are the light quanta. They have the spin 1 and are therefore Bosons. In a Laser the vast majority of emitted photons have the same frequency and propagation direction, they all occupy the same quantum state and form a coherent wave.

The above discussed properties of Bosons and Fermions can be combined into the dictribution functions, they will be treated in the section after the nex section. To understand distribution functions it is necessary to introduce the free energy and the associated chemical potential. Therefore the next section deals with the question:

What are the free energy and the chemical potential?

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