A new form to construct complex superpotentials that produce real energy spectra in supersymmetric quantum mechanics is presented. This is based on the relation between the nonlinear Ermakov equation and a second order differential equation of the Schrödinger type. The superpotentials so constructed are characterized by the Ermakov parameters in such a way that they are always complex-valued and lead to non-Hermitian Hamiltonians with real spectra, whose eigenfunctions form a bi-orthogonal system. As applications we present new complex supersymmetric partners of the free particle that are PT -symmetric and can be either periodic or regular (of the Pöschl-Teller form). A new family of complex oscillators with real frequencies that have the energies of the harmonic oscillator plus an additional real eigenvalue is introduced.
As is known, the Schrodinger equation for a particle in the ring-shaped potential V(r.8) =( 2 4 r -&r2 sin2t9)co, defined in the whole space, has been solved exactly. Here the eigenfunctions are represented in a form which is advantageous for concrete evaluations. The spin-orbit interaction energy Ew. in quasirelativistic approximation is determined analytically, for the first time with a nonspherically symmetric potential. The influence of spin-orbit interaction on the eigenvalues of the spin-free problem and on the selection rules for electrical dipole transitions are investigated as well as the dependence of EL^. on the position and depth of the potential minimum. The model can be useful for investigations of axial symmetric subjects like the benzene molecule or related problems and may be easily extended to a many-electron theory.* Dedicated to Professor Dr. Willy Hartner on the occasion of his 75th birthday t u = (cry, uy, u z ) , with u, = Pauli spin matrices.
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Please cite this article as: H. Cruz, D. Schuch, O. Castaños, O. Rosas-Ortiz, Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian, Annals of Physics (2015), http://dx.
AbstractThe sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.
Based on the Gaussian wave packet solution for the harmonic oscillator and the corresponding creation and annihilation operators, a generalization is presented that also applies for wave packets with time-dependent width as they occur for systems with different initial conditions, time-dependent frequency or in contact with a dissipative environment. In all these cases the corresponding coherent states, position and momentum uncertainties and quantum mechanical energy contributions can be obtained in the same form if the creation and annihilation operators are expressed in terms of a complex variable that fulfills a nonlinear Riccati equation which determines the time-evolution of the wave packet width. The solutions of this Riccati equation depend on the physical system under consideration and on the (complex) initial conditions and have close formal similarities with general superpotentials leading to isospectral potentials in supersymmetric quantum mechanics. The definition of the generalized creation and annihilation operator is also in agreement with a factorization of the operator corresponding to the Ermakov invariant that exists in all cases considered.
Based on three principles taken from empiricism we establish a new nonlinear field theory which allows us to describe frictional effects in dissipative systems with the aid of a Schrodinger-type field equation with logarithmic nonlinearity. This nonlinear field equation corresponds to the classical Langevin equation and can be interpreted in different ways, taking the view of classical undulatory theory or probabilistic theories, respectively. Because of formal similarities, calculations can be performed independently of the accepted interpretation; only the occurring quantities have to be provided with the corresponding meaning. As an example, the nonlinear field equation for the damped harmonic oscillator is solved exactly. The solutions, a wavefunction and a wavepacketlike solution, a solution with Gaussian shape, exhibit reasonable properties, contain the correct reduced frequency f1 = (UJ6 -y2/4)1/2 and are different from the solutions of the undamped problem. The properties of our nonlinear friction term are discussed and compared with those of similar nonlinear friction terms used by other authors. Finally, we derive a nonlinear friction term equivalent to our logarithmic nonlinearity but involving only combinations of suitably defined operators of position and linear momentum and mean values of these operators.
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