Two dimensional non-Hermitian harmonic oscillator: coherent states (original) (raw)
Related papers
An Introduction to Complex Potentials in Quantum Mechanics
Considering a particle moving in a general, three-dimensional complex potential, we show the following: - The anti-Hermitian term of the Hamiltonian, i.e. the imaginary part of the potential, destroys the time-independence of the state-vector norm, which becomes time-dependent, in general. - The total probability, which is equal to the square of the state-vector norm, is also time-dependent, and thus it is not conserved, in general. - The continuity equation, which, in position space, expresses the time derivative of the probability to find the particle in an arbitrary closed region, i.e. in a region with a closed surface as its boundary, contains an extra term, which is proportional to the imaginary part of the potential. - The integral of the extra term in the whole space is equal to the time derivative of the total probability. Thus, the extra term expresses the time derivative of the local probability creation or annihilation which, as we show, behaves exponentially (exponential growth or decay). - In the last two sections, we examine the parity-time reversal transformation (PT-transformation) and we show that the energies of a PT-symmetric potential, i.e. of a complex potential that remains unchanged (invariant) under the action of a PT-transformation, are real if the respective energy eigenstates are also eigenstates of the parity-time reversal operator.
Laser Physics - LASER PHYS, 2007
We provide a reviewlike introduction to the quantum mechanical formalism related to non-Hermitian Hamiltonian systems with real eigenvalues. Starting with the time-independent framework, we explain how to determine an appropriate domain of a non-Hermitian Hamiltonian and pay particular attention to the role played by PJ symmetry and pseudo-Hermiticity. We discuss the time evolution of such systems having in particular the question in mind of how to couple consistently an electric field to pseudo-Hermitian Hamiltonians. We illustrate the general formalism with three explicit examples: (i) the generalized Swanson Hamiltonians, which constitute non-Hermitian extensions of anharmonic oscillators, (ii) the spiked harmonic oscillator, which exhibits explicit super-symmetry, and (iii) the −x 4-potential, which serves as a toy model for the quantum field theoretical ϕ4-theory.
Non-Hermitian Oscillator-Like Hamiltonians and lambda-Coherent States Revisited
Mod Phys Lett a, 2001
Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of self-adjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.
Non-Hermitian Oscillator-Like Hamiltonians and Λ-Coherent States Revisited
Modern Physics Letters A, 2001
Previous λ-deformed non-Hermitian Hamiltonians with respect to the usual scalar product of Hilbert spaces dealing with harmonic oscillator-like developments are (re)considered with respect to a new scalar product in order to take into account their property of selfadjointness. The corresponding deformed λ-states lead to new families of coherent states according to the DOCS, AOCS and MUCS points of view.
New features of scattering from a one-dimensional non-Hermitian (complex) potential
Journal of Physics A: Mathematical and Theoretical, 2012
For complex one-dimensional potentials, we propose the asymmetry of both reflectivity and transmitivity under time-reversal: R(−k) = R(k) and T (−k) = T (k), unless the potentials are real or PT-symmetric. For complex PT-symmetric scattering potentials, we propose that R lef t (−k) = R right (k) and T (−k) = T (k). So far, the spectral singularities (SS) of a onedimensional non-Hermitian scattering potential are witnessed/conjectured to be at most one. We present a new non-Hermitian parametrization of Scarf II potential to reveal its four new features. Firstly, it displays the just acclaimed (in)variances. Secondly, it can support two spectral singularities at two pre-assigned real energies (E * = α 2 , β 2) either in T (k) or in T (−k), when αβ > 0. Thirdly, when αβ < 0 it possesses one SS in T (k) and the other in T (−k). Fourthly, when the potential becomes PT-symmetric [(α + β) = 0], we get T (k) = T (−k), it possesses a unique SS at E = α 2 in both T (−k) and T (k). Lastly, for completeness, when α = iγ and β = iδ, there are no SS, instead we get two negative energies −γ 2 and −δ 2 of the complex PT-symmetric Scarf II belonging to the two well-known branches of discrete bound state eigenvalues and no spectral singularity exists in this case. We find them as E + M = −(γ − M) 2 and E − N = −(δ − N) 2 ; M (N) = 0, 1, 2, ... with 0 ≤ M (N) < γ(δ).
Journal of Physics A: …, 2010
We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian: H = p 2 /2m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h. η turns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, ℑ(ζ +) = −ℑ(ζ −). This in particular contains the PT-symmetric case: ζ + = ζ * −. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of PT-symmetry.
2016
We present a generalization of the perturbative construction of the metric operator for non-Hermitian Hamiltonians with more than one perturbation parameter. We use this method to study the non-Hermitian scattering Hamiltonian: H = p 2 /2m + ζ − δ(x + α) + ζ + δ(x − α), where ζ ± and α are respectively complex and real parameters and δ(x) is the Dirac delta function. For regions in the space of coupling constants ζ ± where H is quasi-Hermitian and there are no complex bound states or spectral singularities, we construct a (positive-definite) metric operator η and the corresponding equivalent Hermitian Hamiltonian h. η turns out to be a (perturbatively) bounded operator for the cases that the imaginary part of the coupling constants have opposite sign, ℑ(ζ +) = −ℑ(ζ −). This in particular contains the PT-symmetric case: ζ + = ζ * −. We also calculate the energy expectation values for certain Gaussian wave packets to study the nonlocal nature of h or equivalently the non-Hermitian nature of H. We show that these physical quantities are not directly sensitive to the presence of PT-symmetry.
The harmonic oscillator as a tutorial introduction to quantum mechanics
2018
Stemming from the similar linearities of the Schrödinger equation in quantum mechanics on the one hand and of the harmonic oscillations in classical mechanics on the other hand, the idea that any N-degree-of-freedom harmonic oscillator (HON) is formally equivalent to a N-level quantum system is put forward. It is shown that the complex dynamic variables α introduced by R. J. Glauber can be regarded as the components of a state vector belonging to some N-dimension complex Hilbert space, and whose time-evolution is ruled by a Schrödinger-like equation. In case the classical HON is parametrically excited, the unitarity of the time-evolution of the associated quantum system is related to the Ehrenfest adiabaticity of the parametric excitation.
IntechOopen, 2023
In the current work, we extend and incorporate the five-axioms probability system of Andrey Nikolaevich Kolmogorov, set up in 1933 the imaginary set of numbers, and this by adding three supplementary axioms. Consequently, any stochastic experiment can thus be achieved in the extended complex probabilities set C which is the sum of the real probabilities set R and the imaginary probabilities set M. The purpose here is to evaluate the complex probabilities by considering additional novel imaginary dimensions to the experiment occurring in the “real” laboratory. Therefore, the random phenomenon outcome and result in C = R + M can be predicted absolutely and perfectly no matter what the random distribution of the input variable in R is since the associated probability in the entire set C is constantly and permanently equal to one. Thus, the following consequence indicates that chance and randomness in R are replaced now by absolute and total determinism in C as a result of subtracting from the degree of our knowledge of the chaotic factor in the probabilistic experiment. Moreover, I will apply to the established theory of quantum mechanics my original complex probability paradigm (CPP) in order to express the quantum mechanics problem considered here completely deterministically in the universe of probabilities C = R + M.
Solution of Schrödinger Equation for Two-Dimensional Complex Quartic Potentials
Communications in Theoretical Physics, 2009
We investigate the quasi-exact solutions of the Schrödinger wave equation for two-dimensional non-hermitian complex Hamiltonian systems within the frame work of an extended complex phase space characterized by x = x 1 + ip 3 , y = x 2 + ip 4 , p x = p 1 + ix 3 , p y = p 2 + ix 4 . 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.