Multiple Scales in Solid State Physics

The quest for an accurate simulations of the physical world, most vividly expressed in the vision of Laplace's daemon [1], is almost as old as quantitative science. Naturally, such a simulation requires the knowledge of all the relevant physical laws, i.e., a Theory of Everything. For the phenomena involving scales larger than an atomic nucleus and smaller than a star, or, equivalently, for processes at ordinary energies, it is known. This Theory of almost Everything is the combination of Newtonian gravity, Maxwell's theory of electrodynamics, Boltzmann's statistical mechanics, and quantum mechanics [2, 3]. Consequently, already shortly after the formulation of the Schrödinger equation, Dirac remarked that the theory behind atomic and solid-state physics, as well as chemistry are completely known [4]. The fundamental equation to be solved for describing the properties of atoms, molecules, or solids is the innocently looking eigenvalue problem

$$H\left| \Psi \right\rangle = E\left| \Psi \right\rangle $$

((1))

where the Hamiltonian for a set of atomic nuclei and their electrons is given by $H = \frac{1}{{2m}}\mathop \sum \limits_j \nabla _j^2 - \mathop \sum \limits_j \frac{1}{{2M_\alpha }}\nabla _\alpha ^2 - \mathop \sum \limits_{\alpha ,j} \frac{{Z_\alpha e^2 }}{{\left| {r_j - R_\alpha } \right|}} + \mathop \sum \limits_{j < k} \frac{{e^2 }}{{\left| {r_j - r_k } \right|}} + \mathop \sum \limits_{\alpha < \beta } \frac{{Z_\alpha Z_\beta e^2 }}{{\left| {r_\alpha - R_\beta } \right|}}.$ Here Z _α and M _α are the atomic number and mass of the α^th nucleus, R _α is its location, e and m are the charge and mass of the electron, and r _j is the location of the j ^th electron. This equation, augmented by gravitational potentials, and including relativistic corrections as the microscopic basis of magnetism, account for the phenomena of our everyday experience. In addition it accounts for quite counterintuitive phenomena, the most spectacular perhaps being macroscopic quantum states like superconductivity, or the entangled states that enable quantum computing.