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Journal of Quantum Information Science

Quantum Mechanics of Complex Octic Potential in One Dimension

Ram Mehar Singh

Published Jan 1, 2013• 0 citations
Publisher

Abstract

For gaining further insight into the nature of the eigenspectra of a complex octic potential [say], we investigate the quasi exact solutions of the Schr?dinger equation in an extended complex phase space characterized by . The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunction of a system. Explicit expressions of eigenvalues and eigenfunctions for the ground state as well as for the first excited state of a complex octic potential and its variant are worked out. It is found that imaginary part of the eigenvalue turns out to be zero for real coupling parameters, whereas it becomes non-zero for complex coupling parameters. However, the PT-symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue even if coupling parameters in the potential are complex.

DOI
10.4236/jqis.2013.31009

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Within the framework of an extended complex phase space approach characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , we investigate the quasi-exact solutions of an analogous Schrödinger equation for a variety of complex sextic potentials of the type V ( x ) = a + bx + cx 2 + dx 3 + ex 4 + fx 5 + hx 6 with complex couplings. It is found that the eigenvalue spectrum in general is complex for complex couplings and real for the real parameters of the potential. While the ground state solutions for an even power sextic potential with an inverse harmonic term and for the - symmetric version of V ( x ) are obtained, other variants of V ( x ) are also investigated with reference to the excited states.
The study of the non-Hermitian Hamiltonian systems within the framework of Supersymmetric Quantum mechanics (SUSY QM) has been carried out in literature in the last decade. Most of these potentials are either PT Symmetric or quasi-Hermitian in nature. But no formulation is advanced in literature to the study of general complex potentials within its framework. In this paper, using the complex phase space approach, we carry out the complexification of Supersymmetric Quantum Mechanics to enable the study of any general non-Hermitian Hamiltonian system within its framework. The commutation relations among its operators are enumerated and the underlying symmetry is discussed. The eigenvalues and eigenfunctions of the real and imaginary parts of superpartner Hamiltonians are also investigated.
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