Complex Phases in Quantum Mechanics

R. S. Kaushal

Aishwarya Parthasarathi, Dayanand Parashar, R. S. Kaushal

With a view to increasing the scope of applications of supersymmetric quantum mechanics (SUSY QM), we formulate the same in a complex phase space. Within this framework, the concept of shape invariance is reinvestigated and an insight into the eigenvalue spectra of non-Hermitian Hamiltonians is sought. The results are applied to a variety of potentials. We claim that the shape invariance for these potentials in the complex phase space can be retrieved in terms of the prescriptions already proposed in the conventional SUSY QM, in that the transformation of potential parameters takes the form of a reflection in the parameter space. Interestingly, some of these features turn out to be the generalization of the concept of quasi-parity used recently in the context of SUSY QM of PT-symmetric potentials. Further, a correspondence of the present approach with other complex formulations of SUSY QM and also in two real dimensions is demonstrated.

Kohji KUMAKURA

Sergio Giardino

Abstract We study the elastic scattering of quantum particles based on a real Hilbert space approach to quaternionic quantum mechanics ( QM) and derive expression for the wave function, the phase shifts, as well as the optical theorem for the case of a hard sphere scattering potential. The strong agreement between these new quaternionic results and the corresponding results in complex quantum mechanics reinforce the validity of the QM generalization of ordinary complex quantum mechanics ( QM).

Slobodan Prvanović

Kerson Huang

A fundamental difference between classical and quantum mechanics is that, while classical mechanics uses real numbers, quantum mechanics uses complex numbers. This is so because quantum mechanics describes a system in terms of wave functions, which are complex numbers, with a modulus and a phase. The phase gives rise to the phenomenon of quantum interference, which has no classical counterpart.