A Linear equation for the Phase Amplitude description of a wave function

George H. Rawitscher

Dieter Schuch

Quantum mechanics is essentially described in terms of complex quantities like wave functions. The interesting point is that phase and amplitude of the complex wave function are not independent of each other, but coupled by some kind of conservation law. This coupling exists in time-independent quantum mechanics and has a counterpart in its time-dependent form. It can be traced back to a reformulation of quantum mechanics in terms of nonlinear real Ermakov equations or equivalent complex nonlinear Riccati equations, where the quadratic term in the latter equation explains the origin of the phase-amplitude coupling. Since realistic physical systems are always in contact with some kind of environment this aspect is also taken into account. In this context, different approaches for describing open quantum systems, particularly effective ones, are discussed and compared. Certain kinds of nonlinear modifications of the Schrödinger equation are discussed as well as their interrelations and their relations to linear approaches via non-unitary transformations. The modifications of the aforementioned Ermakov and Riccati equations when environmental effects are included can be determined in the time-dependent case. From formal similarities conclusions can be drawn how the equations of time-independent quantum mechanics can be modified to also incluce the enviromental aspects.

George H. Rawitscher, Victo S. Filho, Thiago Carvalho Peixoto

Maxi San Miguel

The interplay between the polarization state of light and transverse effects in lasers is analyzed through an amplitude equation description of an atomic transition between spin sublevels. Linearly polarized traveling waves are found, whose stability is restricted by a phase instability associated with the direction of polarization. The instability persists for polarization stabilized lasers. Novel states of laser light such as standing waves with a spatially periodic linear polarization or coexisting traveling waves with different wave numbers and circular polarizations are also found.

George H. Rawitscher

V.V. Babikov