The behavior of the solutions of the SchrSdinger (Helmholtz) equation in more than one dimension in a classically forbidden region has been investigated in numerous studies in quantum chemistry, the theory of atomic collisions, nuclear theory, the theory of diffraction, and so forth.

Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.Classical mechanics is a singular theory in that real-energy classical particles can never enter classically forbidden regions. However, if one regulates classical mechanics by allowing the energy E of a particle to be complex, the particle exhibits quantum-like behavior: complex-energy classical particles can travel between classically allowed regions separated by potential barriers. When Im E → 0, the classical tunneling probabilities persist. Hence, one can interpret quantum tunneling as an anomaly. A numerical comparison of complex classical tunneling probabilities with quantum tunneling probabilities leads to the conjecture that as Re E increases, complex classical tunneling probabilities approach the corresponding quantum probabilities. Thus, this work attempts to generalize the Bohr correspondence principle from classically allowed to classically forbidden regions.

arXiv:1011.0121Journal of Physics A Mathematical and TheoreticalAug 23, 201126 citations

Solutions of Schrödinger’s Equation in One Dimension

Bruce Cameron Reed

Springer eBooksJan 1, 20220 citations

Solution of the Schrödinger Equation in One Dimension

S. Brandt, Hans Dieter Dahmen

Springer eBooksJan 1, 20010 citations

"Proud waves" and wave-mechanical trajectories

A. Orefice, Raffaele Giovanelli, Domenico Ditto

We present here the exact, trajectory based wave-mechanical description of particle dynamics provided by the time-independent Schroedinger equation, of which the time-dependent equation allows a statistical treatment. Let us remind, for comparison, that Bohmian mechanics is largely recognized to provide a basically hydrodynamic formulation of the quantum theory, describing the evolution of physical systems in terms of statistical streamlines which indicate the regions of space where the moving particles preferentially gather: the very concept of true particle path is avoided, in Bohmian theory, by a weak, non-perturbing measurement. In the present paper the wave-dynamical Hamiltonian equations providing the exact particle trajectories are directly obtained from the time-independent Schroedinger equation, and numerical solutions are presented for a number of different external potentials acting on the moving particles. A Wave-Potential function (holding for any - even classical - wave process described by a Helmholtz-like equation) is shown to be the one and only cause of wave-like features such as diffraction and interference, and to act, for matter waves, without any wave-particle energy exchange.We present here the exact, trajectory based wave-mechanical description of particle dynamics provided by the time-independent Schroedinger equation, of which the time-dependent equation allows a statistical treatment. Let us remind, for comparison, that Bohmian mechanics is largely recognized to provide a basically hydrodynamic formulation of the quantum theory, describing the evolution of physical systems in terms of statistical streamlines which indicate the regions of space where the moving particles preferentially gather: the very concept of true particle path is avoided, in Bohmian theory, by a weak, non-perturbing measurement. In the present paper the wave-dynamical Hamiltonian equations providing the exact particle trajectories are directly obtained from the time-independent Schroedinger equation, and numerical solutions are presented for a number of different external potentials acting on the moving particles. A Wave-Potential function (holding for any - even classical - wave process described by a Helmholtz-like equation) is shown to be the one and only cause of wave-like features such as diffraction and interference, and to act, for matter waves, without any wave-particle energy exchange.

arXiv:1604.05068Apr 18, 20160 citations

Solution of the Schrödinger Equation in One Dimension

S. Brandt, Hans Dieter Dahmen

Springer eBooksJan 1, 19950 citations

Classical interpretation of probability oscillations in low-energy atomic collisions

Pierre Botheron, Bernard Pons

We perform close-coupling molecular and classical trajectory Monte Carlo (CTMC) calculations of charge-exchange probabilities in low-energy ${\mathrm{H}}^{+}$ , ${\mathrm{He}}^{2+}+\mathrm{H}$ collisions. We consider head-on collisions with a zero impact parameter and study the dependence of the charge-exchange probability on the impact velocity. This probability exhibits oscillations generally understood as signatures of quantum phase effects. CTMC calculations with dimensionality reduced to one fairly reproduce the oscillations and show that these latter stem from the back and forth motion of the electron between the nuclear centers in the internuclear region. In three dimensions, the probability oscillations are generally washed out in the statistical CTMC framework because of the electron transverse degrees of freedom. However three-dimensional Bohmian trajectories ascertain the classical interpretation of the oscillations in a quantum framework.We perform close-coupling molecular and classical trajectory Monte Carlo (CTMC) calculations of charge-exchange probabilities in low-energy ${\mathrm{H}}^{+}$, ${\mathrm{He}}^{2+}+\mathrm{H}$ collisions. We consider head-on collisions with a zero impact parameter and study the dependence of the charge-exchange probability on the impact velocity. This probability exhibits oscillations generally understood as signatures of quantum phase effects. CTMC calculations with dimensionality reduced to one fairly reproduce the oscillations and show that these latter stem from the back and forth motion of the electron between the nuclear centers in the internuclear region. In three dimensions, the probability oscillations are generally washed out in the statistical CTMC framework because of the electron transverse degrees of freedom. However three-dimensional Bohmian trajectories ascertain the classical interpretation of the oscillations in a quantum framework.

Physical Review AJun 15, 20119 citations

We perform close-coupling molecular and classical trajectory Monte Carlo (CTMC) calculations of charge-exchange probabilities in low-energy

{\mathrm{H}}^{+}

{\mathrm{He}}^{2+}+\mathrm{H}

collisions. We consider head-on collisions with a zero impact parameter and study the dependence of the charge-exchange probability on the impact velocity. This probability exhibits oscillations generally understood as signatures of quantum phase effects. CTMC calculations with dimensionality reduced to one fairly reproduce the oscillations and show that these latter stem from the back and forth motion of the electron between the nuclear centers in the internuclear region. In three dimensions, the probability oscillations are generally washed out in the statistical CTMC framework because of the electron transverse degrees of freedom. However three-dimensional Bohmian trajectories ascertain the classical interpretation of the oscillations in a quantum framework.