## ABSTRACT

In this chapter, the statistical mechanics equations of the ideal gas are derived from thermodynamic derivatives of the free energy; the average energy is also calculated as a statistical average, and the chemical potential for an ideal gas is introduced. The harmonic oscillator is discussed next on three levels: (1) the macroscopic oscillator and (2) the microscopic oscillator treated by classical statistical mechanics, which behaves adequately at high temperature (leading to the experimental Dulong-Petit law), but misbehaves at low T. (3) The quantum mechanical oscillator, which leads correctly to the behavior of the entropy and the heat capacity at low temperature, S → 0 and C_{V}
→ 0 as T → 0. The correspondence principle of quantum mechanics is satisfied, as at high T the quantum equations lead to the classical ones, S = 3Nk
_{B}[1 + ln(hν/k
_{B}
T)] and C_{V }
= 3Nk
_{B}. Comparing the properties of these oscillators (e.g., the constant amplitude of a macroscopic oscillator versus the average amplitude of a microscopic oscillator that can be exceeded) deepens the understanding of the probabilistic nature of statistical mechanics.