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Atoms

Solving Time-Dependent Schödinger Equation for Some PT-Symmetric Quantum Mechanical Problems

Tsin-Fu Jiang

Published Sep 11, 2024• 0 citations
Publisher

Abstract

Using a high-precision code, we generate the eigenstates of a PT-symmetric Hamiltonian. We solve the time-dependent Schrödinger equation (TDSE) of the non-Hermitian system based on the eigenset. Since the formulation is relatively new and the observables are calculated differently than conventional quantum mechanics, we justify it with a paradigmatic case in Hermitian quantum mechanics. We present the harmonic generation spectra on some model PT-Hamiltonians driven by an electric pulse. We discuss the physical differences with the harmonic spectra of a pulse-driven atom.

DOI
10.3390/atoms12090046

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Using a high-precision code, we generate the eigenstates of a PT-symmetric Hamiltonian. We solve the time-dependent Schrödinger equation (TDSE) of the non-Hermitian system based on the eigenset. Since the formulation is relatively new and the observables are calculated differently than conventional quantum mechanics, we justify it with a paradigmatic case in Hermitian quantum mechanics. We present the harmonic generation spectra on some model PT-Hamiltonians driven by an electric pulse. We discuss the physical differences with the harmonic spectra of a pulse-driven atom.
We study in this paper the time evolution of PT-symmetric non-Hermitian Hamiltonian consisting of periodically driven SU(1,1) generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a PT-symmetric but non-unitary transformation operator in the construction of the non-Hermitian invariant. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time- evolution operator is found analytically. As a consequence of the non-unitarity the ket (|ψ(t)〉) and bra (〈ψ(t)|) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.
We derive a new time-dependent Schrödinger equation(TDSE) for quantum models with non-hermitian Hamiltonian. Within our theory, the TDSE is symmetric in the two Hilbert spaces spanned by the left and the right eigenstates, respectively. The physical quantities are also identical in these two spaces. Based on this TDSE, we show that exchanging two quasi-particles in a non-hermitian model can generate arbitrary geometric phase. The system can also violate the Lieb-Robinson bound in non-relativistic quantum mechanics so that an action in one place will immediately cause a change in the distance. We show that the above two surprising behaviors can also appear in anyonic model, which makes us propose that the non-hermitian single particle model may possess many common features with anyonic model.
Recent work in mathematical physics and nonlinear optics has shown that Hamiltonians that are non-Hermitian but still symmetric under parity and time reversal can describe eigenstates of a system with real eigenvalues. Other research has also showed that the nonlinear Schrodinger equation can be generalized to describe PT-symmetric systems, which generates novel solutions not described by its Hermitian equivalent. The Hermitian form of the nonlinear Schroedinger equation can also be extended to describe a particular case of the general PT-symmetric NLS, suggesting a connection between the two. I attempted to generate a unitary operator that will be useful for unitary quantum algorithms describing a coupled set of nonlinear Schroedinger equations and the PT-symmetric version of the NLS.
Time-dependent P\mathcal {P}PT\mathcal {T}T-symmetric quantum mechanics
The parity-time-reversal ( )-symmetric quantum mechanics (QM) (PTQM) has developed into a noteworthy area of research. However, to date, most known studies of PTQM focused on the spectral properties of non-Hermitian Hamiltonian operators. In this work, we propose an axiom in PTQM in order to study general time-dependent problems in PTQM, e.g., those with a time-dependent -symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a time-dependent Schrödinger-like equation of motion for PTQM to the familiar time-dependent Schrödinger equation in conventional QM. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional QM. Under our proposed framework, we further study in detail the Berry-phase generation in a class of -symmetric two-level systems. It is found that a closed path in the parameter space of PTQM is often associated with an open path in a properly mapped problem in conventional QM. In one interesting case, we further interpret the Berry phase as the flux of a continuously tunable fictitious magnetic monopole, thus highlighting the difference between PTQM and conventional QM despite the existence of a proper mapping between them.
In this report we examine the concept of PT -symmetric quantum mechanics and analyze two such systems. PT -symmetric quantum mechanics is a new branch of quantum mechanics, first proposed by Bender and Boettcher in 1998 and has been the subject of much research since then. Various important aspects of PT -symmetric quantum mechanics is explained, such as asymptotic behaviour of solutions of the Schrödinger equation, as well as the concept of the C operator that is unique to PT -symmetric quantum mechanics. We briefly discuss both the concept of broken and unbroken PT -symmetry, and the relation between PT -symmetric and Hermitian quantum mechanics through similarity transforms. We specifically direct our attention to two theoretical quantum mechanical systems described by Hamiltonians with the property of being PT -symmetric. For these systems we determine the energy levels, one by analytical means and the other by use of numerical methods. To determine the energy levels numerically we have used the builtin ODE-solver ode45 in MATLAB in conjunction with the equation solver fzero. Our results are compared to those of previous studies and our methods and results are discussed.
In the context of non-Hermitian quantum mechanics, many systems are known to possess a pseudo PT symmetry , i.e. the non-Hermitian Hamiltonian H is related to its adjoint H^{†} via the relation, H^{†}=PTHPT . We propose a derivation of pseudo PT symmetry and η -pseudo-Hermiticity simultaneously for the time dependent non-Hermitian Hamiltonians by intoducing a new metric η(t)=PTη(t) that not satisfy the time-dependent quasi-Hermiticity relation but obeys the Heisenberg evolution equation. Here, we solve the SU(1,1) time-dependent non-Hermitian Hamiltonian and we construct a time-dependent solutions by employing this new metric and discuss a concrete physical applications of our results.