M. Masili, ICTP

This lesson starts by the study of the internal degrees of freedom, examples are the electronic, the nuclear, the vibrational and the rotational *etc*.. The Hamiltonian will be then the sum of the Hamiltonians corresponding to each degree of freedom and the same for the **specific heat**. Then the lesson focused in the study of **Ideal quantum gases** in particular an ideal **gas of bosons ****Ideal Bose **gas and ideal** gas of fermions Fermi gases. **

**The thermodynamics of the ideal ****Bose gas** was studied with the help of the **grand partition function **and the

**equation of state**as well as the number of particles where calculated as a function of the

**thermal de Broglie wavelength**and the

**Fugacity**. It was shown that in order to have a positive number of particles in the lowest energy level the

**chemical potential**(?) have to be smaller than the lowest energy level in this case ? is negative (?<0). Then both cases were considered the high and low temperature regime. It was demonstrated that in the high temperature regime the

**specific heat**for the ideal

**Bose gas**is larger than the

**Ideal gas**converging to it from above for a very large temperature, on the other hand we know that at T=0 it has to go to zero thus a maximum in the

**specific heat**was predicted. For very low temperature it was obtained that most of particles will occupy the ground state, this phenomenon is known as the

**Bose–Einstein condensate**.

The critical temperature at which the **Bose–Einstein condensate** take place was calculated and the physical properties of the condensate studied, for example it was shown that this gas has infinite **compressibility**, the **specific heat** will goes to zero as T^{3/2} as the particles that are in the ground state does not contribute to the **specific heat** because they cannot transfer energy, they are in a zero energy state, and this is why the **specific heat** goes to zero.

As an example of this ideal **Bose gas** it was considered a system composed by **photons**. This system can be consider as a system of particles called **photons** occupying the energy levels of a **harmonic oscillator **and ?=0, the expressions for the calculation of the average number of particles and average energy for each possible frequency ? were obtained. The internal energy U is found to be U~T^{4}, the **equation of state** is given by PV = U/3 and the **entropy** is S=4U/3T ~ T^{3} .