PhD thesis in English

Chapter 1Introduction1.1 ForewordThe essential ingredients of the quantum mechanical theory are dual particle-wavenature of the matter, and the notion of identical particles. The quantum indistinguishabilityof particles has a profound impact on the statistical properties. Whileparticles with a half-integer spin (fermions) try to avoid each other due to the Pauliexclusionprinciple, particles with an integer spin (bosons) do not exhibit such restrictionsand are actually, as a consequence of the minimal energy principle, tryingto occupy the same single-particle ground state. These effects are captured by twodifferent probability distributions that describe the thermal equilibrium in the twotypes of physical systems. The Fermi-Dirac distribution for fermions was originallyderived in 1926 by its two authors independently, while trying to introduce quantizationconcepts into an ideal gas of particles obeying the Pauli exclusion principle.The Bose-Einstein distribution for bosons was introduced in 1924 by the joint effortof Bose and Einstein [1, 2]. At the time, it was a missing piece of knowledge for thecomplete explanation of the Planck’s law of black body radiation.In the high-temperature limit, both distributions are well approximated by acommon Maxwell-Boltzmann distribution. The condition necessary for the effectsof the quantum statistics to become observable can be roughly estimated as follows.The noninteracting particles of the mass M affect each other if their thermal deBroglie wavelength λ T at temperature T,√2πλ T =2k B TM , (1.1)is comparable to the inter-particle distance given by n −1/3 , where n is a typical particledensity. Naturally, the notion of the “high temperature” and “low temperature”depends on the features of the considered system. For example, electrons in a typical1