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Journal of Physics A Mathematical and Theoretical

The finite P T -symmetric square well potential

GÉZA LÉVAI, József Kovács

Published Nov 9, 2018• 3 citations
Publisher

Abstract

Abstract The -symmetric version of the one-dimensional finite square well potential is constructed by supplementing the real finite square well potential with constant imaginary components outside the well. This study is motivated by the unusual features of the -symmetric Rosen–Morse II potential, which has an imaginary component with similar asymptotics. The exponentially vanishing (i.e. normalizable) solutions are constructed, after determining the corresponding energy eigenvalues from the zeros of a transcendental equation. It is found that only real energy eigenvalues are allowed, similarly to the case of the Rosen–Morse II potential. Transmission and reflection coefficients are determined and are found to exhibit handedness. It is shown that due to the non-vanishing imaginary potential component, bound states correspond to the zeros of the reflection coefficient. Similarities and differences with respect to the real finite square well and the -symmetric Rosen–Morse II potential are discussed.

DOI
10.1088/1751-8121/aaefb0

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Abstract The -symmetric version of the one-dimensional finite square well potential is constructed by supplementing the real finite square well potential with constant imaginary components outside the well. This study is motivated by the unusual features of the -symmetric Rosen–Morse II potential, which has an imaginary component with similar asymptotics. The exponentially vanishing (i.e. normalizable) solutions are constructed, after determining the corresponding energy eigenvalues from the zeros of a transcendental equation. It is found that only real energy eigenvalues are allowed, similarly to the case of the Rosen–Morse II potential. Transmission and reflection coefficients are determined and are found to exhibit handedness. It is shown that due to the non-vanishing imaginary potential component, bound states correspond to the zeros of the reflection coefficient. Similarities and differences with respect to the real finite square well and the -symmetric Rosen–Morse II potential are discussed.
Two PT-symmetric potentials are compared, which possess asymptotically finite imaginary components: the PT-symmetric Rosen-Morse II and the finite PT-symmetric square well potentials. Despite their different mathematical structure, their shape is rather similar, and this fact leads to similarities in their physical characteristics. Their bound-state energy spectrum was found to be purely real, an this finding was attributed to their asymptotically non-vanishing imaginary potential components. Here the V(x)= γδ(x)+ i2Λ sgn(x) potential is discussed, which can be obtained as the common limit of the two other potentials. The energy spectrum, the bound-state wave functions and the transmission and reflection coefficients are studied in the respective limits, and the results are compared.
Bound and scattering solutions of the -symmetric Rosen–Morse II potential are investigated. The energy eigenvalues and the corresponding wavefunctions are written in a closed analytic form, and it is shown that this potential always supports at least one bound state. It is found that with increasing non-Hermiticity the real bound-state energy spectrum does not turn into complex conjugate pairs, i.e. the spontaneous breakdown of symmetry does not occur, rather the energy eigenvalues remain real and shift to positive values. Closed expression is found for the pseudo-norm of the bound states, and its sign is found to follow the (−1)n rule. Similarly to the known scattering examples, the reflection coefficients exhibit a handedness effect, while the transmission coefficient picks up a complex phase factor when the direction of the incoming wave is reversed. It is argued that the unusual findings might be caused by the asymptotically non-vanishing, though finite imaginary potential component. Comparison with the real Rosen–Morse II potential is also made.
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.
Abstract The real exact solutions to the relativistic spinless particle with modified Rosen–Morse potential are presented. The novel issue is that the previously obtained eigenfunctions related to this type of potential in an imaginary variable i sinh (α x) (Dirac case) can be expressed by the real Nomanovski polynomials. The energy levels are calculated numerically. It is interesting to note that the energy levels first increase with the parameter V1 (V1<0) and then decrease with it. In the case of the 𝒫𝒯 symmetric version it is found that the eigenvalues are real even though the 𝒫𝒯 symmetric potentials are non‐Hermitian.
The finite square potential well is a staple problem in introductory quantum mechanics. There is an extensive literature on the determination of the allowed energies, which requires the solution of a transcendental equation by numerical, graphical or approximate analytic methods. Here we investigate the less explored problem of a particle in a semi-infinite potential well. The energy eigenvalues, which are also determined by a transcendental equation, are found by a standard graphical method, and a simple rule that yields the number of stationary states is provided. Next a simplification of the aforementioned transcendental equation is attempted. During the process pitfalls are encountered and a purportedly simpler graphical treatment of the problem given in the solutions manual to a fine textbook is shown to be flawed. A more careful analysis brings forth the correct simplification, which is shown to be particularly suitable for finding highly accurate approximations to the energy levels. Finally, a class of exact solutions is produced, the associated normalized eigenfunctions are constructed and the probability of finding the particle inside the well is computed.