Quantum Mechanical Solution Research Articles

Many-particle Hamiltonian for open systems with full Coulomb interaction: Application to classical and quantum time-dependent simulations of nanoscale electron devices

A many-particle Hamiltonian for a set of particles with Coulomb interaction inside an open system is described without any perturbative or mean-field approximation. The boundary conditions of the Hamiltonian on the borders of the open system [in the real three-dimensional (3D) space representation] are discussed in detail to include the Coulomb interaction between particles inside and outside of the open system. The many-particle Hamiltonian provides the same electrostatic description obtained from the image-charge method, but it has the fundamental advantage that it can be directly implemented into realistic (classical or quantum) electron device simulators via a 3D Poisson solver. Classically, the solution of this many-particle Hamiltonian is obtained via a coupled system of Newton-type equations with a different electric field for each particle. The quantum-mechanical solution of this many-particle Hamiltonian is achieved using the quantum (Bohm) trajectory algorithm [X. Oriols, Phys. Rev. Lett. 98, 066803 (2007)]. The computational viability of the many-particle algorithms to build powerful nanoscale device simulators is explicitly demonstrated for a (classical) double-gate field-effect transistor and a (quantum) resonant tunneling diode. The numerical results are compared with those computed from time-dependent mean-field algorithms showing important quantitative differences.

Vibrational Relaxation of OH and CH Fundamentals of Polar and Nonpolar Molecules in the Condensed Phase

Studies of vibrational energy flow in various polar and nonpolar molecules that follows the ultrafast excitation of the CH and OH stretch fundamentals, modeled using semiclassical methods, are reviewed. Relaxation rates are calculated using Landau-Teller theory and a time-dependent method, both of which consider a quantum mechanical solute molecule coupled to a classical bath of solvent molecules. A wide range of decay rates are observed, ranging from 1 ps for neat methanol to 50 ps for neat bromoform. In order to understand the flow rates, it is argued that an understanding of the subtle mixing between the solute eigenstates is needed and that solute anharmonicities are critical to facilitating condensed phase vibrational relaxation. The solvent-assisted shifts of the solute vibrational energy levels are seen to play a critical role of enhancing or decreasing lifetimes.

Exact classical quantum mechanical solution for atomic helium which predicts conjugate parameters from a unique solution for the first time

Quantum mechanics (QM) and quantum electrodynamics (QED) are often touted as the most successful theories ever. In this paper, this claim is critically evaluated by a test of internal consistency for the ability to calculate the conjugate observables of the nature of the free electron, ionization energy, elastic electron scattering, and the excited states of the helium atom using the same solution for each of the separate experimental measurements. It is found that in some cases quantum gives good numbers, but the solutions are meaningless numbers since each has no relationship to providing an accurate physical model. Rather, the goal is to mathematically reproduce an experimental or prior theoretical number using adjustable parameters including arbitrary wave functions in computer algorithms with precision that is often much greater (e.g. 8 significant figures greater) than possible based on the propagation of errors in the measured fundamental constants implicit in the physical problem. Given the constraints of adherence to physical laws and internal consistency, an extensive literature search indicates that quantum mechanics has never solved a single physical problem correctly including the hydrogen atom and the next member of the periodic chart, the helium atom. Rather than using postulated unverifiable theories that treat atomic particles as if they were not real, physical laws are now applied to the same problem. In an attempt to provide some physical insight into atomic problems and starting with the same essential physics as Bohr of − e moving in the Coulombic field of the proton and the wave equation as modified after Schrodinger, a classical approach is explored which yields a model which is remarkably accurate and provides insight into physics on the atomic level. The proverbial view deeply seated in the wave-particle duality notion that there is no large-scale physical counterpart to the nature of the electron is shown not to be correct. Physical laws and intuition may be restored when dealing with the wave equation and quantum atomic problems. Specifically, a theory of classical quantum mechanics (CQM) was derived from first principles as reported previously [1-7] that successfully applied physical laws to the solution of atomic problems that has its basis in a breakthrough in the understanding of the stability of the bound electron to radiation. Rather than using the postulated Schrodinger boundary condition: “ 0 Ψ→ as →∞ r ,” which leads to a purely mathematical model of the electron, the constraint is based on experimental observation. Using Maxwell’s equations, the classical wave equation is solved with the constraint that the bound 1 = n -state electron cannot radiate energy. Although it is well known that an accelerated point particle radiates, an extended distribution, modeled as a superposition of accelerating charges, does not have to radiate. A simple invariant physical model arises naturally wherein the predicted results are extremely straightforward and internally consistent requiring minimal math as in the case of the most famous equations of Newton, Maxwell, Einstein, de Broglie, and Planck on which the model is based. No new physics is needed; only the known physical laws based on direct observation are used. The accurate solution of the helium atom is confirmed by the agreement of predicted and observed conjugate parameters using the same unique physical model in all cases.

A study of errors in trajectory simulation with relevance for 0.2–50 eV electrons in liquid water

A highly simplified model of the elastic scattering of electrons in a nm-size volume of liquid water, where the water molecules are regarded as point scatterers and inelastic scattering is neglected, is studied for electron energies 0.2–50 eV. This model allows an exact quantum mechanical solution of the multiple elastic scattering problem. The exact solutions are compared with the corresponding trajectory simulations. Two different data sets for the elastic scattering mean free path are discussed and used, one based on cross sections for water molecules in the gas phase and the other on cross sections derived from scattering in amorphous ice. In each case, the comparison provides a detailed insight into the character and magnitude of the error in the trajectory simulation, and gives a preliminary indication of an upper bound to the limits of validity of trajectory simulation in the real case of electron scattering in liquid water. Main results: with a fully random distribution of scatterers, the trajectory simulation is found to be a surprisingly good approximation down to quite low electron energies ( ∼ 10 eV ). Below a few eV, the error increases rapidly with decreasing electron energy. A substantial increase of the error in trajectory simulation results when short-range order (minimum distance between scatterers) is introduced; the observed effect may partly be understood from the theory of kinematic diffraction in liquids. At very low electron energies (a few eV) one may note a quantum effect in the form of a standing wave pattern in the average distribution of scattering events.

Solution of the master equation for Wigner’s quasiprobability distribution in phase space for the Brownian motion of a particle in a double well potential

Quantum effects in the Brownian motion of a particle in the symmetric double well potential V(x)=ax(2)2+bx(4)4 are treated using the semiclassical master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p). The equilibrium position autocorrelation function, dynamic susceptibility, and escape rate are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded has a quantum correction depending strongly on the barrier height and is compared with that given analytically by the quantum mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution. Moreover, the low-frequency part of the spectrum associated with noise assisted Kramers transitions across the potential barrier may be accurately described by a single Lorentzian with characteristic frequency given by the quantum mechanical reaction rate.

Simulating water with the self-consistent-charge density functional tight binding method: from molecular clusters to the liquid state.

The recently developed self-consistent-charge density functional tight binding (SCCDFTB) method provides an accurate and inexpensive quantum mechanical solution to many molecular systems of interests. To examine the performance of the SCCDFTB method on (liquid) water, the most fundamental yet indispensable molecule in biological systems, we report here the simulation results of water in sizes ranging from molecular clusters to the liquid state. The latter simulation was achieved through the use of the linear scaling divide-and-conquer approach. The results of liquid water simulation indicate that the SCCDFTB method can describe the structural and energetics of liquid water in qualitative agreement with experiments, and the results for water clusters suggest potential future improvements of the SCCDFTB method.

Semiclassical master equation in Wigners phase space applied to Brownian motion in a periodic potential

The quantum Brownian motion of a particle in a cosine periodic potential V(x)= -V{0}cos(x/x{0}) is treated using the master equation for the time evolution of the Wigner distribution function W(x,p,t) in phase space (x,p) . The dynamic structure factor, escape rate, and jump-length probabilities are evaluated via matrix continued fractions in the manner customarily used for the classical Fokker-Planck equation. The escape rate so yielded is compared with that given analytically by the quantum-mechanical reaction rate solution of the Kramers turnover problem. The matrix continued fraction solution substantially agrees with the analytic solution.

Scattering at the intersections of potentials

The scattering process influenced by the intersection of potential energy surfaces is studied. Exact quantum-mechanical solutions to the two-dimensional two- and three-state models have been found for the collision energy equal to the potential energy at the intersection point. The models considered are the Jahn-Teller and Renner-Teller intersections and the triple intersection of the double cone with a plane. Closed analytical expressions for the scattering amplitudes have been obtained. The results reveal the geometric phase effect and the nonadiabatic phase impact on the differential cross section.

Quantum calculation of vortices in the inner crust of neutron stars

The self-consistent mean-field quantum mechanical solution of a vortex and a nucleus immersed in a sea of free neutrons, a scenario representative of the inner crust of neutron stars, is presented for the first time. Because of quantal size effects the phase space for vortices inside the nucleus is essentially zero, so that the vortex core opens up and surrounds the nucleus. As a consequence, pinned configurations (in which a vortex becomes anchored to the nucleus) are favored at low and high densities in the inner crust. This result is qualitatively different from that obtained in all previous models, which predict pinning at intermediate densities.

Combined Multireference Configuration Interaction/ Molecular Dynamics Approach for Calculating Solvatochromic Shifts: Application to thenO→ π* Electronic Transition of Formaldehyde

A combined quantum mechanics/molecular mechanics method is described here for considering the solvatochromic shift of excited states in solution. The quantum mechanical solute is described using high level multireference configuration interaction methods (MRCI), while molecular dynamics is used for obtaining the structure of the solvent around the solute. The electrostatic effect of the solvent is included in the quantum description of the solute in an averaged way. This method is used to study solvent effects on the n(O) --> pi* electronic transition of formaldehyde in aqueous solution. The effects of solute polarization, basis sets, and dynamical correlation on the solvatochromic shift, and on dipole moments, have been investigated.

Projections of Quantum Observables onto Classical Degrees of Freedom in Mixed Quantum-Classical Simulations: Understanding Linear Response Failure for the Photoexcited Hydrated Electron

We present a general analytic method for understanding how specific motions of a classical bath influence the dynamics of quantum-mechanical observables in mixed quantum-classical molecular dynamics simulations. We apply our method and develop expressions for the special case of quantum solvation, allowing us to examine how specific classical solvent motions couple to the equilibrium energy fluctuations and nonequilibrium energy relaxation of a quantum-mechanical solute. As a first application of our formalism, we investigate the motions of classical water underlying the equilibrium and nonequilibrium excited-state solvent response functions of the hydrated electron; the results allow us to explain why the linear response approximation fails for this system.

Dynamics of Nonadiabatic Chemical Reactions

New methods are proposed to treat nonadiabatic chemical dynamics in realistic large molecular systems by using the Zhu-Nakamura (ZN) theory of curve-crossing problems. They include the incorporation of the ZN formulas into the Herman-Kluk type semiclassical wave packet propagation method and the trajectory surface hopping (TSH) method, formulation of the nonadiabatic transition state theory, and its application to the electron transfer problem. Because the nonadiabatic coupling is a vector in multidimensional space, the one-dimensional ZN theory works all right. Even the classically forbidden transitions can be correctly treated by the ZN formulas. In the case of electron transfer, a new formula that can improve the celebrated Marcus theory in the case of normal regime is obtained so that it can work nicely in the intermediate and strong electronic coupling regimes. All these formulations mentioned above are demonstrated to work well in comparison with the exact quantum mechanical numerical solutions and are expected to be applicable to large systems that cannot be treated quantum mechanically numerically exactly. To take into account another quantum mechanical effect, namely, the tunneling effect, an efficient method to detect caustics from which tunneling trajectories emanate is proposed. All the works reported here are the results of recent activities carried out in the author's research group. Finally, the whole set of ZN formulas is presented in Appendix.

Quantum theory of an atom laser originating from a Bose-Einstein condensate or a Fermi gas in the presence of gravity

We present a three-dimensional quantum-mechanical theory of radio-frequency-outcoupled atom lasers from trapped atomic gases in the presence of the gravitational force. Predictions for the total outcoupling rate as a function of the radio frequency and for the beam wave function are given. We establish a sum rule for the energy-integrated outcoupling, which leads to a separate determination of the coupling strength between the atoms and radiation field. For a noninteracting Bose-Einstein condensate analytic solutions are derived which are subsequently extended to include the effects of atomic interactions. The interactions enhance interference effects in the beam profile and modify the outcoupling rate of the atom laser. We provide a complete quantum-mechanical solution which is in line with experimental findings and allows us to determine the validity of commonly used approximative methods. We also extend the formalism to a fermionic atom laser and analyze the effect of superfluidity on the outcoupling of atoms.

Exact solution of a quantized LC circuit coupled to a power source

By means of a two-step unitary transformation, the exact quantum mechanical solution of two-dimensional mesoscopic circuit coupled via inductance is derived. The wavefunction, propagator and expectation value of Hamiltonian are evaluated. The fluctuations of charges and currents are calculated and uncertainty products between charges and their conjugate currents are obtained. We apply our results to the system with special cases of the power source such as the sinusoidal and the sawtooth power source and show that the probability densities oscillate with time due to the power source.

Using classical probability functions to illuminate the relation between classical and quantum physics

The classical and quantum mechanical solutions to three simple physical systems are directly compared by using the classical equations of motion to determine a probability density function. This function is compared to the probability density function for the quantum mechanical stationary state with the same energy. The nonclassical behavior of the quantum mechanical systems is thus made more distinct. This comparison allows students to gain experience using probability functions to describe physical systems and to see how the quantum mechanical solutions reduce to classical behavior in the high-energy/short-wavelength limit.

Problems and solutions in quantum mechanics

Problems and solutions in quantum mechanics

Nonadiabatic transmission: Exact quantum-mechanical solution for a special case of the two-state exponential model

The exact quantum-mechanical solution of the two-state exponential model with one flat diabatic potential has been found for a specific energy below the higher threshold. The $N$ matrix has been derived in the adiabatic representation and the closed analytical expression for the transmission coefficient has been obtained. The results reveal the influence of the closed channel on the nonadiabatic transmission.

ELECTRON TRANSFER RATE UNIFORMLY VALID FROM NONADIABATIC TO ADIABATIC REGIME BASED ON THE ZHU–NAKAMURA THEORY

On the basis of the generalized nonadiabatic transition state theory recently introduced to remedy the crucial deficiencies of the conventional transition state theory, we have presented a new formula for electron transfer rate, which can cover the whole range from adiabatic to nonadiabatic regime in the absence of solvent dynamics control. The rate is expressed as a product of the well-known Marcus theory and a new coefficient that represents the effects of nonadiabatic transition at the crossing seam surface. The numerical comparisons are performed with different approaches and the present approach shows an excellent agreement with the quantum mechanical numerical solutions from weak to strong electronic coupling. The explanation of the experimental data of Nelsen et al. manifests the potential applicability of the present theory.

TIME-DEPENDENT 2D SPACETIMES FROM MATRICES

Solutions of matrix quantum mechanics have been shown to describe time dependent backgrounds in the holographically dual two-dimensional closed string theory. We review some recent work dealing with nontrivial spacetimes which arise in this fashion and discuss aspects of physical phenomena in them. These include spacetimes with spacelike boundaries.

Bound states in Andreev billiards with soft walls

The energy spectrum and the eigenstates of a rectangular quantum dot containing soft potential walls in contact with a superconductor are calculated by solving the Bogoliubov--de Gennes equation. We compare the quantum mechanical solutions with a semiclassical analysis using a Bohr-Sommerfeld (BS) quantization of periodic orbits. We propose a simple extension of the BS approximation which is well suited to describe Andreev billiards with parabolic potential walls. The underlying classical periodic electron-hole orbits are directly identified in terms of ``scar''-like features engraved in the quantum wave functions of Andreev states which we determine here explicitly.