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arXiv:2012.04526 · Physica Scripta

Two-dimensional non-Hermitian harmonic oscillator: coherent states

Masoumeh Izadparast, S. Habib Mazharimousavi

Published May 23, 2019• 2 citations
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Abstract

Abstract In this study, we introduce a two-dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time-independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.

DOI
10.1088/1402-4896/ab240b

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Abstract In this study, we introduce a two-dimensional complex harmonic oscillator potential with space and time reflection symmetries. The corresponding time-independent Schrödinger equation yields real eigenvalues with complex eigenfunctions. We also construct the coherent state of the system by using a superposition of 12 eigenfunctions. Using the complex correspondence principle for the probability density we investigate the possible modifications in the probability densities due to the non-Hermitian aspect of the Hamiltonian.
We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.
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.
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.
A quantum-mechanical model of a damped harmonic oscillator (both with time-independent and time-dependent parameters) is studied in the framework of the linear Schr\"odinger equation with a Hermitian nonstationary Hamiltonian. Integrals of the motion of this equation and their eigenstates, including coherent states, are constructed. The influence of an external harmonic force to the time evolution of various average values calculated over coherent states is considered, including the resonance case. The specific symmetry of the Hamiltonian leading to the new concept of loss-energy states is discussed.