Brief on quantum mechanics and Schrödinger operators

Carlos Miguel Madrid Casado, Collado Villalba

The aim of this paper is to give a brief account of the development of the mathematical equivalence of quantum mechanics. In order to deal with atomic systems, Heisenberg developed matrix mechanics in 1925. Some time later, in the winter of 1926, Schrodinger established his wave mechanics. In the spring of 1926, quantum physicists had two theoretical models that allowed them to predict the same behaviour of the quantum systems, but both of them were very different. Schrodinger thought that the empirical equivalence could be explained by means of a proof of mathematical equivalence.

акустика Механика

Jochen Autschbach

Abstract Introduction of the postulates of quantum mechanics: Wavefunctions, operators, observables, commutating operators, expectation values, probabilities, Heisenberg uncertainty. The postulates are then used to set up a ‘quantum recipe’, i.e. a straightforward recipe by which to write down the (nonrelativistic) quantum Hamiltonian of a system of particles. This chapter also discusses the representation of quantum operators as matrices, in reference to a set of ‘basis’ functions, and the variation principle. The idea of a particle trajectory must be abandoned in quantum mechanics. Observable properties of a particle correspond to eigenvalues of the associated quantum operators. The chapter concludes with a brief discussion of the Schrodinger’s cat paradox, quantum entanglement, and other oddities.

In this work we present a brief review about quantum mechanics in phase space. The approach discussed is based in the notion of symplectic structure and star-operators. In this sense, unitary representations for the Galilei group are construct, and the Schrodinger equation in phase space is derived. The connection between phase space amplitudes and Wigner function is presented. As a new result we solved the Schrodinger equation in phase space for simple pendulum. PACS Numbers: 11.10.Nx, 11.30.Cp, 05.20.Dd

Nima Moshayedi