Schr¨Odinger Representation of Quantum Mechanics, Berry Connection, and Superconductivity

Hiroyasu Koizumi

G. X. A. Petronilo, S. C. Ulhoa, A. E. Santana

We explore the concept of the extended Galilei group, a representation for the symplectic quantum mechanics in the manifold G, written in the light-cone of a five-dimensional de Sitter space-time in the phase space. The Hilbert space is constructed endowed with a symplectic structure. We study the unitary operators describing rotations and translations, whose generators satisfy the Lie algebra of G. This representation gives rise to the Schr¨odinger (Klein–Gordon-like) equation for the wave function in the phase space such that the dependent variables have the position and linear momentum contents. The wave functions are associated to the Wigner function through the Moyal product such that the wave functions represent a quasiamplitude of probability. We construct the Pauli–Schr¨odinger (Dirac-like) equation in the phase space in its explicitly covariant form. Finally, we show the equivalence between the five-dimensional formalism of the phase space with the usual formalism, proposing a solution that recovers the non-covariant form of the Pauli–Schr¨odinger equation in the phase space.

D. E. Thomsen

Herbert E. Spohn, Stefan Teufel

We reconsider the time-dependent Born-Oppenheimer theory with the goal to carefully separate between the adiabatic decoupling of a given group of energy bands from their orthogonal subspace and the semiclassics within the energy bands. Band crossings are allowed and our results are local in the sense that they hold up to the rst time when a band crossing is encountered. The adiabatic decoupling leads to an eective Schrodinger equation for the nuclei, including contributions from the Berry connection. 1

Nikolay Leonidovich Chuprikov

Using the one-dimensional Schr\{o}dinger equation as an example, it is shown that for any self-adjoint operator it is possible to uniquely predict the value of the corresponding observable (including momentum, kinetic and total energy of the particle) at that spatial point where the particle will be accidentally detected at a given instant of time. It is shown that the Schr\{o}dinger formalism ensures the fulfillment of the relations of wave-particle duality, which connect the energy and momentum of a particle with the frequency and wave number of the wave function, for any spatial point and any instant of time.