Reconstructing interference fringes in slit experiments by complex quantum trajectories

There are external and internal representations for a quantum state Ψ. External representation is commonly adopted in the standard quantum mechanics by exploiting probability density function Ψ*Ψ to explain the observed interference fringes in slit experiments. On the other hand, in quantum Hamilton mechanics, the quantum state Ψ has a dynamical representation that reveals the internal mechanism underlying the externally observed interference fringes. The internal representation of Ψ is described by a set of Hamilton equations of motion, by which quantum trajectories of a particle moving in Ψ can be solved. In this article, millions of complex quantum trajectory connecting slits to a screen are solved from the Hamilton equations, and the statistical distribution of their arrivals on the screen is shown to reproduce the observed interference fringes. This appears to be the first quantitative verification of the equivalence between the trajectory‐based statistics and the wavefunction‐based statistics on slit experiments. © 2012 Wiley Periodicals, Inc.     

Theory for non-equilibrium statistical mechanics

This paper reviews a new theory for non-equilibrium statistical mechanics. This gives the non-equilibrium analogue of the Boltzmann probability distribution, and the generalization of entropy to dynamic states. It is shown that this so-called second entropy is maximized in the steady state, in contrast to the rate of production of the conventional entropy, which is not an extremum. The relationships of the new theory to Onsager's regression hypothesis, Prigogine's minimal entropy production theorem, the Langevin equation, the formula of Green and Kubo, the Kawasaki distribution, and the non-equilibrium fluctuation and work theorems, are discussed. The theory is worked through in full detail for the case of steady heat flow down an imposed temperature gradient. A Monte Carlo algorithm based upon the steady state probability density is summarized, and results for the thermal conductivity of a Lennard-Jones fluid are shown to be in agreement with known values. Also discussed is the generalization to non-equilibrium mechanical work, and to non-equilibrium quantum statistical mechanics. As examples of the new theory two general applications are briefly explored: a non-equilibrium version of the second law of thermodynamics, and the origin and evolution of life.     

粒子隧道几率的一种新形式

基于二阶线型微分方程的多个任意阶转向点，建立了粒子进入或跳出势垒或势井时状态波函数的连结公式，导出了粒子穿过势垒的隧道几率公式，并利用严格的数学方法讨论了粒子进入势井时的量子化条件。该公式可合理地还原到一阶转向点情况，并与早期利用一阶转向点近似所推导出的公式相一致。     

Arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential

Within the framework of supersymmetric quantum mechanics method, we study by an algebraic method the arbitrary l‐wave bound states of the Schrödinger equation for the hyperbolical molecular potential by a proper approximation to nonlinear centrifugal term. The explicitly analytical formula of energy levels is derived, and the corresponding bound state wave functions are presented. The function analysis method is used to rederive the same energy levels of the quantum system under consideration to check the validity of this algebraic method. In addition, it is shown from numerical results of energy levels that above certain α parameter depending on the choices of potential parameters V₁ and V₂ the hyperbolical molecular potential cannot trap a particle as it becomes weaker and the energy level starts to turn positive when the potential parameter α becomes larger. © 2014 Wiley Periodicals, Inc.     

On the hydrogen atom beyond the Born–Oppenheimer approximation

Recently a new formulation of quantum mechanics has been suggested which is based on the concept of signed particles, that is, classical objects provided with a position, a momentum and a sign simultaneously. In this article, we comment on the plausibility of simulating atomic systems beyond the Born–Oppenheimer approximation by means of the signed particle formulation of quantum mechanics. First, to show the new perspective offered by this new formalism, we provide an example studying quantum tunnelling through a simple Gaussian barrier in terms of the signed particle formulation. Then, we perform a direct simulation of the hydrogen atom as a full quantum two‐body system, showing that the formalism can be a very promising tool for first‐principle‐only quantum chemistry.     

Wilson–Sommerfeld quantization rule revisited

A fresh look at the origin of the Wilson–Sommerfeld quantization rule has been pursued to gain new insight. The rule is shown to provide states that satisfy several well‐known theorems of standard quantum mechanics. A few other useful results and scaling relations are also derived. They emerge to act as nice guiding rules of thumb in the course of rigorous computations. Certain features of true excited‐state densities can be understood. Goodness of approximate densities can be assessed. Compressed systems can be studied profitably. A route is also sketched that allows one to retrieve classical trajectories from near‐exact energy eigenfunctions for both bound and resonant states by exploiting this rule. Additionally, a discussion on semiclassical perturbation theory is presented emphasizing the asymptotic behavior. Pilot calculations demonstrate the success of the present endeavor under various circumstances. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 113–125, 2001     

Negative Mass Can Be Positively Useful in Quantum Mechanics

We show that in nonrelativistic quantum mechanics, the concept of negative mass can be utilized to solve Schrödinger equations and increase the convergence rates of the basis set expansion solutions for quantum eigenvalue problems. In particular, when a negative‐mass particle moves under a repulsive interaction, the state of motion turns out to be bound at a positive energy. This counterintuitive behavior can be employed to deal with physical systems where repulsive interactions strongly perturb the stable states established by the competing attractive interactions such as many‐electron atoms. Helium‐like atoms are used to illustrate the solution scheme.     

Quantum molecular mechanics—a noniterative procedure for the fast Ab Initio calculation of closed shell systems

In this article, we advance the foundations of a strategy to develop a molecular mechanics method based not on classical mechanics and force fields but entirely on quantum mechanics and localized electron‐pair orbitals, which we call quantum molecular mechanics (QMM). Accordingly, we introduce a new manner of calculating Hartree–Fock ab initio wavefunctions of closed shell systems based on variationally preoptimized nonorthogonal electron pair orbitals constructed by linear combinations of basis functions centered on the atoms. QMM is noniterative and requires only one extremely fast inversion of a single sparse matrix to arrive to the one‐particle density matrix, to the electron density, and consequently, to the ab initio electrostatic potential around the molecular system, or cluster of molecules. Although QMM neglects the smaller polarization effects due to intermolecular interactions, it fully takes into consideration polarization effects due to the much stronger intramolecular geometry distortions. For the case of methane, we show that QMM was able to reproduce satisfactorily the energetics and polarization effects of all distortions of the molecule along the nine normal modes of vibration, well beyond the harmonic region. We present the first practical applications of the QMM method by examining, in detail, the cases of clusters of helium atoms, hydrogen molecules, methane molecules, as well as one molecule of HeH⁺ surrounded by several methane molecules. We finally advance and discuss the potentialities of an exact formula to compute the QMM total energy, in which only two center integrals are involved, provided that the fully optimized electron‐pair orbitals are known. © 2012 Wiley Periodicals, Inc.     

Statistical mechanical theory for steady state systems. V. Nonequilibrium probability density

The phase space probability density for steady heat flow is given. This generalizes the Boltzmann distribution to a nonequilibrium system. The expression includes the nonequilibrium partition function, which is a generating function for statistical averages and which can be related to a nonequilibrium free energy. The probability density is shown to give the Green-Kubo formula in the linear regime. A Monte Carlo algorithm is developed based upon a Metropolis sampling of the probability distribution using an umbrella weight. The nonequilibrium simulation scheme is shown to be much more efficient for the thermal conductivity of a Lennard-Jones fluid than the Green-Kubo equilibrium fluctuation method. The theory for heat flow is generalized to give the generic nonequilibrium probability densities for hydrodynamic transport, for time-dependent mechanical work, and for nonequilibrium quantum statistical mechanics.     

Classical and quantum chaos in molecular rotational excitation by ac electric fields

The interaction of diatomic molecules with an ac electric field is described by a periodically driven rigid rotor model Hamiltonian. Numerical studies of the classical and quantum dynamics reveal a remarkably close correspondence between classically chaotic dynamics and quantum time evolution. Unlike the periodically kicked rotor all the quasienergy states located in the (bounded) chaotic region in phase space are extended states. Expanded in the free rotor basis, their coefficients fullfill the statistical predictions for random vectors. Consequently, even in off resonance condition the probability for transfering angular momentum to the diatomic molecule is large and eventually the firstj _m excited rotational states will be “democratically” populated. The value ofj _m is determined by the bounded chaotic region in phase space. The rotational occupation probability shows an erratic behavior with fluctuations following the statistical predictions for random quantum states.     

Computation of Free Energy

Many quantities that are standardly used to characterize a chemical system are related to free‐energy differences between particular states of the system. By statistical mechanics, free‐energy differences may be expressed in terms of averages over ensembles of atomic configurations for the molecular system of interest. Here, we review the most useful formulae to calculate free‐energy differences from ensembles generated by molecular simulation, illustrate a number of recent developments, and highlight practical aspects of such calculations with examples selected from the literature.     

Quantum trajectories for resonant scattering

Time‐dependent wave packet resonant scattering for the double square barrier has been studied in terms of Bohm quantum trajectories. The high transmission probability for the wave packet with a resonant energy can be explained by the behavior of the quantum trajectories under the influence of the relatively slow formation of a node within the first barrier. This node splits the trajectories into reflected and transmitted components. During this stage, many particle trajectories pass through the double‐barrier region and contribute to the transmitted part of the wave packet. Due to the transient nature of the nodes, trajectories in the reflected wave packet bunch together between the nodes for a finite period of time so that temporary structure (localization of particles and accompanying increase in the probability density) develops on small length scales. These calculations also show that the particles gain high momentum near the nodal points, and they reach a uniform momentum distribution after transmitting the barrier region. We have found that the presence of a node between the two barriers influences the different lifetimes of the quasi‐bound states. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 81: 206–213, 2001     

Electrically Controlled Eight‐Spin‐Qubit Entangled‐State Generation in a Molecular Break Junction

The generation of spin‐based multi‐qubit entangled states in the presence of an electric field is one of the most challenging tasks in current quantum‐computing research. Such examples are still elusive. By using non‐equilibrium Green′s function‐based quantum‐transport calculations in combination with non‐collinear spin density functional theory, we report that an eight‐spin‐qubit entangled state can be generated with the high‐spin state of a dinuclear Fe(II) complex when the system is placed in a molecular break junction. The possible gate operation scheme, gating time, and decoherence issues have been carefully addressed. Furthermore, our calculations reveal that the preservation of the high spin state of this complex is possible if the experimentalists keep the electric‐field strength below 0.78 V nm^?1. In brief, the present study offers a unique way to realize the first example of a multi‐qubit entangled state by electrical means only.     

Synthesizing quantum probability by a single chaotic complex‐valued trajectory

The current trajectory interpretation of quantum mechanics is based on an ensemble viewpoint that the evolution of an ensemble of Bohmian trajectories guided by the same wavefunction Ψ converges asymptotically to the quantum probability . Instead of the Bohm's ensemble‐trajectory interpretation, the present paper gives a single‐trajectory interpretation of quantum mechanics by showing that the distribution of a single chaotic complex‐valued trajectory is enough to synthesize the quantum probability. A chaotic complex‐valued trajectory manifests both space‐filling (ergodic) and ensemble features. The space‐filling feature endows a chaotic trajectory with an invariant statistical distribution, while the ensemble feature enables a complex‐valued trajectory to envelop the motion of an ensemble of real trajectories. The comparison between complex‐valued and real‐valued Bohmian trajectories shows that without the participation of its imaginary part, a single real‐valued trajectory loses the ensemble information contained in the wavefunction Ψ, and this explains the reason why we have to employ an ensemble of real‐valued Bohmian trajectories to recover the quantum probability . © 2015 Wiley Periodicals, Inc.     

Machine learning and signed particles,an alternative and efficient way to simulate quantum systems

Recently neural networks have been applied in the context of the signed particle formulation of quantum mechanics to rapidly and reliably compute the Wigner kernel of any provided potential. Important advantages were introduced, such as the reduction of the amount of memory required for the simulation of a quantum system by avoiding the storage of the kernel in a multi-dimensional array, as well as attainment of consistent speedup by the ability to realize the computation only on the cells occupied by signed particles. An inherent limitation was the number of hidden neurons to be equal to the number of cells of the discretized real space. In this work, anew network architecture is presented, decreasing the number of neurons in its hidden layer, thereby reducing the complexity of the network and achieving an additional speedup. The approach is validated on a onedimensional quantum system consisting of a Gaussian wave packet interacting with a potential barrier.     

Matula numbers, Gödel numbering and Fock space

By making use of Matula numbers, which give a 1-1 correspondence between rooted trees and natural numbers, and a Gödel type relabelling of quantum states, we construct a bijection between rooted trees and vectors in the Fock space. As a by product of the aforementioned correspondence (rooted trees $\leftrightarrow $ ? Fock space) we show that the fundamental theorem of arithmetic is related to the grafting operator, a basic construction in many Hopf algebras. Also, we introduce the Heisenberg–Weyl algebra built in the vector space of rooted trees rather than the usual Fock space. This work is a cross-fertilization of concepts from combinatorics (Matula numbers), number theory (Gödel numbering) and quantum mechanics (Fock space).     

Quantum state transformation by optimal projective measurements

This paper explores the optimal control of quantum state transformations in finite-dimensional quantum systems by a sequence of non-selective projective measurements. In our schemes, the projectors of each measurement are represented by a unitary matrix. Through variational analysis of the objective function over the unitary group, the necessary condition for a measurement sequence to be a critical point of the underlying state transformation objective is found to be a highly symmetric form as a chain of equalities. Since these equality relations are generally difficult to solve analytically, we focus on the fundamental case employing a single measurement, in which analytical solutions for maximizing the state transformation probability are found between pure states, or between mixed and pure states, or between orthogonal mixed states under two typical type of measurements. These results suggest a new way of designing optimal quantum dynamics control strategies by quantum measurements.