Michele Cini

Grand Canonical Ensemble

A gas is in contact with a surface. On the surface we find N 0 localized and distinguishable sites adsorbing N (N≤N 0) molecules of the gas (each site can adsorb zero or one molecule of the gas). Find the grand canonical partition function of the system, and determine the chemical potential as a function of the average number of particles 〈N〉 which are adsorbed by the surface. You can think that the canonical partition function of an adsorbed molecule is a function only of the temperature, Q(T), and that all the adsorbed molecules are non interacting.

Systems of Particles

We need to extend the quantum mechanical theory to the case of N particles; separately, the particles would be described by a Schrodinger equation or the Pauli equation, depending on their spins. As in the classical case, the Hamiltonian of the system will be the sum of those of the particles plus an interaction term (possibly). For independent particles (no interaction), the Hamiltonian is, in obvious notation

Quantum Statistical Mechanics

\hat{H}(1,2,\ldots , N)= \sum _{n}^N \hat{h}(n),

Thermodynamics and Microcanonical Ensemble

where h(n) describes particle n. The wave function \(\varPsi (1,2,\ldots , N)\) depends on all the orbital and spin degrees of freedom of the particles, so it is a spinor in the spin space of each particle.

Formalism of Quantum Mechanics and One Dimensional Problems

Let us start with the Heavyside (Oliver Heaviside (1850–1925) was probably the first to use the \(\delta \) before Dirac, and the work of George Green also implies the concept. Often the names are not historically fair) \(\theta \) discontinuous function, also known as the step function, defined by

Summary of Quantum and Statistical Mechanics

\begin{aligned} \theta (x) = \left\{ \begin{array}{ll} 1 &{} \text {se } x>0, \\ {1 \over 2} &{} \text {if } x=0, \\ 0 &{} \text {se } x<0, \end{array} \right. \end{aligned}