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International Journal of Theoretical Physics

Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and $$(1/\hat x)\hat p + \hat p(1/\hat x)$$ Terms

Jeong Ryeol Choi

Published Apr 1, 2003• 17 citations
Publisher
DOI
10.1023/a:1024423018984

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Journal of the Korean Physical Society•Feb 1, 2016•4 citations
Exact Wave Functions of Time-Dependent Hamiltonian Systems Involving Quadratic, Inverse Quadratic, and (1/x^)p^+p^(1/x^)(1/\hat x)\hat p + \hat p(1/\hat x)(1/x^)p^​+p^​(1/x^) Terms
Comparison of Corrected Wave Functions Associated to Two Different Approaches for the Time-Dependent Hamiltonian Systems Involving (1/x^)p^+p^(1/x^)(1/\hat{x})\hat{p}+\hat{p}(1/\hat{x})(1/x^)p^​+p^​(1/x^) Term
Using an algebraic approach, it is possible to obtain the temporal evolution wave function for a Gaussian wave-packet obeying the quadratic time-dependent Hamiltonian (QTDH). However, in general, most of the practical cases are not exactly solvable, for we need general solutions of the Riccatti equations which are not generally known. We therefore bypass directly solving for the temporal evolution wave function, and study its inverse problem. We start with a particular evolution of the wave-packet, and get the required Hamiltonian by using the inverse method. The inverse approach opens up a new way to find new exact solutions to the QTDH. Some typical examples are studied in detail. For a specific time-dependent periodic harmonic oscillator, the Berry phase is obtained exactly.
Quantum solutions of a time-dependent Hamiltonian for the motion of a time-varying mass subjected to time-dependent singular potentials in three dimensions are investigated. A time-dependent inverse quadratic potential and a Coulomb-like potential are considered as the components of the singular potential of the system. Because the Hamiltonian is a function of time, special techniques for deriving quantum solutions of the system are necessary. A quadratic invariant operator is introduced, and its eigenstates are derived using the Nikiforov-Uvarov method together with a unitary transformation method. The Nikiforov-Uvarov method enables us to solve the eigenvalue equations of the invariant operator, which are second-order linear diffierential equations, by reducing the original equation to a hypergeometric type. According to the invariant operator theory, the wave functions of the system are represented in terms of the eigenstates obtained in such a way. The difference of the wave functions from the eigenstates of the invariant operator is that the wave functions have time-dependent phases while the eigenstates do not. By determining the phases of the wave functions via the help of the Schr¨odinger equation, we identify the full wave functions of the system and address their physical implications.