a matrix that describes a quantum system in a mixed state, a statistical ensemble of several quantum states (here several similar 2-state atoms). This should be contrasted with a single state vector that describes a quantum system in a pure state. The expectation value of a measurement, , over the ensemble is . In terms of a complete set of states, , one can write Von Neumann's equation tells us how the density matrix evolves with time. In equilibrium, one has , and the allowed density matrices are . The canonical ensemble has where . For the 2-state system, we can write . Here is the gyromagnetic ratio. Hence , and From which Explanation of para and diamagnetism using perturbation theory In the presence of a uniform external magnetic field along the z-direction, the Hamiltonian of the atom changes by where are positive real numbers which are independent of which atom we are looking at but depend on the mass and the charge of the electron. corresponds to individual electrons of the atom. We apply second order perturbation theory to this situation. This is justified by the fact that even for highest presently attainable field strengths, the shifts in the energy level due to is quite small w.r.t. atomic excitation energies. Degeneracy of the original Hamiltonian is handled by choosing a basis which diagonalizes in the degenerate subspaces. Let be such a basis for the state of the atom (rather the electrons in the atom). Let be the change in energy in . So we get In our case we can ignore and higher order terms. We get In case of diamagnetic material, the first two terms are absent as they don't have any angular momentum in their ground state. In case of paramagnetic material all the three terms contribute. Adding spin-spin interaction in the Hamiltonian: Ising model So far, we have assumed that the atoms do not interact with each other. Even though this is a reasonable assumption in the case of diamagnetic and paramagnetic substances, this assumption fails in the case of ferromagnetism, where the spins of the atom try to align with each other to the extent permitted by the thermal agitation. In this case, we have to consider the Hamiltonian of the ensemble of the atom. Such a Hamiltonian will contain all the terms described above for individual atoms and terms corresponding to the interaction among