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Physica Scripta

Quantum Mechanics of Complex Sextic Potentials in One Dimension

Aishwarya Parthasarathi, R. S. Kaushal

Published Jan 1, 2003• 19 citations
Publisher

Abstract

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.

DOI
10.1238/physica.regular.068a00115

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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.
With a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one-dimensional Schrödinger equation corresponding to a non-Hermitian Hamiltonian H ( x , p ) we investigate the ground-state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x = x 1 + i p 2 , p = p 1 + i x 2 , where ( x 1 , p 1 ) and ( x 2 , p 2 ) are real and considered as canonical pairs. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems. It is noted that the imaginary part of the eigenvalue, E i , turns out to be zero for all potentials V ( x ) with real couplings whereas it turns out to be nonzero for the case when the couplings are complex. The prescription is also extended to study the excited states. The problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed.
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.
We investigate the quasi-exact solutions of the Schr?dinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x1 + ip3, y = x2 + ip4, px = p1 + ix3, py = p2 + ix4. Explicit expressions of the energy eigenvalues and the eigenfunctions for ground and first excited states for a complex quartic potential are obtained. Eigenvalue spectra of some variants of the complex quartic potential, including PT-symmetric one, are also worked out.
With a view to getting further insight into the solutions of one-dimensional analogous Schr?dinger equation for a non-hermitian (complex) Hamiltonian system, we investigate the quasi-exact PT- symmetric solutions for an octic potential and its variant using extended complex phase space approach characterized by x=x1+ip2, p=p1+ix2, where (x1, p1) and (x2, p2) are real and considered as canonical pairs. Besides the complexity of the phase space, complexity of potential parameters is also considered. The analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalue and eigenfunction of a system. The imaginary part of energy eigenvalue of a non-hermitian Hamiltonian exist for complex potential parameters and reduces to zero for real parameters. However, in the present work, it is found that imaginary component of the energy eigenvalue vanishes even when potential parameters are complex, provided that PT-symmetric condition is satisfied. Thus PT- symmetric version of a non-hermitian Hamiltonian possesses the real eigenvalue.