Quantum Mechanical Probability Research Articles

A study of quantum mechanical probabilities in the classical Hodgkin-Huxley model.

The Hodgkin-Huxley (HH) model is a powerful model to explain different aspects of spike generation in excitable cells. However, the HH model was proposed in 1952 when the real structure of the ion channel was unknown. It is now common knowledge that in many ion-channel proteins the flow of ions through the pore is governed by a gate, comprising a so-called "selectivity filter" inside the ion channel, which can be controlled by electrical interactions. The selectivity filter (SF) is believed to be responsible for the selection and fast conduction of particular ions across the membrane of an excitable cell. Other (generally larger) parts of the molecule such as the pore-domain gate control the access of ions to the channel protein. In fact, two types of gates are considered here for ion channels: the "external gate", which is the voltage sensitive gate, and the "internal gate" which is the selectivity filter gate (SFG). Some quantum effects are expected in the SFG due to its small dimensions, which may play an important role in the operation of an ion channel. Here, we examine parameters in a generalized model of HH to see whether any parameter affects the spike generation. Our results indicate that the previously suggested semi-quantum-classical equation proposed by Bernroider and Summhammer (BS) agrees strongly with the HH equation under different conditions and may even provide a better explanation in some cases. We conclude that the BS model can refine the classical HH model substantially.

Probability Densities of Superconductors

The energy, current density and momentum probability densities of superconductors were studied from London, Ginzburg-Landau and BSC theories by treating cooper pair as a particle moving in a magnetic field through analytical and numerical techniques. The London and GL solution were exactly the same at the classical limit for NbN. Considering a Cooper pair as a complete classical particle, the momentum probability density was derived by using the Maxwell velocity distribution and the quantum mechanical momentum probability density was derived by using the radial wave function of the cooper pairs for Zn. The quantum mechanical and classical momentum probability densities overlap at zero momentum.

Probability Densities of Superconductors

The energy, current density and momentum probability densities of superconductors were studied from London, Ginzburg-Landau and BSC theories by treating cooper pair as a particle moving in a magnetic field through analytical and numerical techniques. The London and GL solution were exactly the same at the classical limit for NbN. Considering a Cooper pair as a complete classical particle, the momentum probability density was derived by using the Maxwell velocity distribution and the quantum mechanical momentum probability density was derived by using the radial wave function of the cooper pairs for Zn. The quantum mechanical and classical momentum probability densities overlap at zero momentum.

Calculation of the Quantum-Mechanical Tunneling in Bound Potentials

The quantum-mechanical tunneling is often important in low-energy reactions, which involve motion of light nuclei, occurring in condensed phase. The potential energy profile for such processes is typically represented as a double-well potential along the reaction coordinate. In a potential of this type defining reaction probabilities, rigorously formulated only for unbound potentials in terms of the scattering states with incoming/outgoing scattering boundary conditions, becomes ambiguous. Based on the analysis of a rectangular double-well potential, a modified expression for the reaction probabilities and rate constants suitable for arbitrary double- (or multiple-) well potentials is developed with the goal of quantifying tunneling. The proposed definition involves energy eigenstates of the bound potential and exact quantum-mechanical transmission probability through the barrier region of the corresponding scattering potential. Applications are given for several model systems, including proton transfer in a HO–H–CH3 model, and the differences between the quantum-mechanical and quasiclassical tunneling probabilities are examined.

Relational causality and classical probability: Grounding quantum phenomenology in a superclassical theory

By introducing the concepts of "superclassicality" and "relational causality", it is shown here that the velocity field emerging from an n-slit system can be calculated as an average classical velocity field with suitable weightings per channel. No deviation from classical probability theory is necessary in order to arrive at the resulting probability distributions. In addition, we can directly show that when translating the thus obtained expression for said velocity field into a more familiar quantum language, one immediately derives the basic postulate of the de Broglie-Bohm theory, i.e. the guidance equation, and, as a corollary, the exact expression for the quantum mechanical probability density current. Some other direct consequences of this result will be discussed, such as an explanation of Born's rule and Sorkin's first and higher order sum rules, respectively.

LINEARITY OF QUANTUM PROBABILITY MEASURE AND HARDY'S MODEL

We re-examine d = 4 hidden-variables model for a system of two spin-1/2 particles in view of the concrete model of Hardy, who analyzed the criterion of entanglement without referring to inequality. The basis of our analysis is the linearity of the probability measure related to the Born probability interpretation, which excludes noncontextual hidden-variables model in d≥3. To be specific, we note the inconsistency of the noncontextual hidden-variables model in d = 4 with the linearity of the quantum mechanical probability measure in the sense 〈ψ|a⋅σ ⊗b ⋅σ|ψ〉+ 〈ψ|a ⋅σ ⊗b′ ⋅σ|ψ〉 = 〈ψ|a⋅σ⊗(b + b′)⋅σ|ψ〉 for noncollinear b and b′. It is then shown that Hardy's model in d = 4 does not lead to a unique mathematical expression in the demonstration of the discrepancy of local realism (hidden-variables model) with entanglement and thus his proof is incomplete. We identify the origin of this nonuniqueness with the nonuniqueness of translating quantum mechanical expressions into expressions in hidden-variables model, which results from the failure of the above linearity of the probability measure. In contrast, if the linearity of the probability measure is strictly imposed, which tantamounts to asking that the noncontextual hidden-variables model in d = 4 gives the Clauser–Horne–Shimony–Holt (CHSH) inequality |〈B〉|≤2 uniquely, it is shown that the hidden-variables model can describe only separable quantum mechanical states; this conclusion is in perfect agreement with the so-called Gisin's theorem which states that |〈B〉|≤2 implies separable states.

Describe Quantum Mechanics in Dual 4 d Complex Space-Time and the Ontological Basis of Wave Function

Micro-object is both particle and wave, so the traditional Particle Model (mass point model) is actually not applicable for it. Here to describe its motion, we expand the definition of time and space and pick up the spatial degrees of freedom hidden by particle model. We say that micro-object is like a rolling field-matter-ball, which has four degrees of freedom including one surface curvature degree and three mapping degrees in the three-dimensional phenomenal space. All the degrees are described by four curvature coordinate components, namely “k1, k2, k3, k4”, which form the imaginary part of a complex phase space, respectively. While as to the real part, we use “x1, x2, x3, x4” to describe the micro object’s position in our real space. Consequently, we build a Dual 4-dimensional complex phase space whose imaginary part is 4-dimension k space and real part is 4-dimension x space to describe the micro-object’s motion. Furthermore, we say that wave function can describe the information of a field-matter-ball’s rotation & motion and also matter-wave can spread the information of micro-object’s spatial structure & density distribution. Matter-wave and probability-wave can transform to each other though matter-wave is a physical wave. The non-point property is the foundational source of the probability in Quantum Mechanics.

Flexoelectricity from density-functional perturbation theory

We derive the complete flexoelectric tensor, including electronic and lattice-mediated effects, of an arbitrary insulator in terms of the microscopic linear response of the crystal to atomic displacements. The basic ingredient, which can be readily calculated from first principles in the framework of density-functional perturbation theory, is the quantum-mechanical probability current response to a long-wavelength acoustic phonon. Its second-order Taylor expansion in the wave vector $\mathbf{q}$ around the $\ensuremath{\Gamma}$ ($\mathbf{q}=0$) point in the Brillouin zone naturally yields the flexoelectric tensor. At order one in $\mathbf{q}$ we recover Martin's theory of piezoelectricity [Martin, Phys. Rev. B 5, 1607 (1972)], thus providing an alternative derivation thereof. To put our derivations on firm theoretical grounds, we perform a thorough analysis of the nonanalytic behavior of the dynamical matrix and other response functions in a vicinity of $\ensuremath{\Gamma}$. Based on this analysis, we find that there is an ambiguity in the specification of the ``zero macroscopic field'' condition in the flexoelectric case; such arbitrariness can be related to an analytic band-structure term, in close analogy to the theory of deformation potentials. As a by-product, we derive a rigorous generalization of the Cochran-Cowley formula [Cochran and Cowley, J. Phys. Chem. Solids $\mathbf{23}$, 447 (1962)] to higher orders in $\mathbf{q}$. This can be of great utility in building reliable atomistic models of electromechanical phenomena, as well as for improving the accuracy of the calculation of phonon dispersion curves. Finally, we discuss the physical interpretation of the various contributions to the flexoelectric response, either in the static or dynamic regime, and we relate our findings to earlier theoretical works on the subject.

(Invited) Si/SiGe Resonant Interband Tunnel Diodes Grown by Large-Area Chemical Vapor Deposition

Si/SiGe resonant interband tunnel diodes were successfully technology transferred from low-temperature molecular beam epitaxial growth to chemical vapor deposition (CVD) on 200-mm diameter p-doped silicon wafers in a production style reactor. The resonant interband tunnel diode structure consists of a p+-i-n+ diode that incorporates vapor phase doped δ-doping to enhance quantum mechanical confinement and tunneling probability. The tunneling barrier thickness was varied from 2 nm to 8 nm and a record high peak-to-valley current ratio of 5.2 for a CVD process is reported for a 6 nm barrier thickness. The CVD reactor is an active integral tool within a 200mm full CMOS process line and is suggestive that Si/SiGe RITDs could be seamlessly integrated into a full hybrid CMOS process flow.

Prethermalization revealed by the relaxation dynamics of full distribution functions

We detail the experimental observation of the non-equilibrium many-body phenomenon prethermalization. We study the dynamics of a rapidly and coherently split one-dimensional Bose gas. An analysis based on the use of full quantum mechanical probability distributions of matter wave interference contrast reveals that the system evolves toward a quasi-steady state. This state, which can be characterized by an effective temperature, is not the final thermal equilibrium state. We compare the evolution of the system to an integrable Tomonaga–Luttinger liquid model, and show that the system dephases to a prethermalized state rather than undergoing thermalization toward a final thermal equilibrium state.

A numerical method for the fractional Schrödinger type equation of spatial dimension two

Abstract This work focuses on an investigation of the (n+1)-dimensional time-dependent fractional Schrödinger type equation. In the early part of the paper, the wave function is obtained using Laplace and Fourier transform methods and a symbolic operational form of the solutions in terms of Mittag-Leffler functions is provided. We present an expression for the wave function and for the quantum mechanical probability density. We introduce a numerical method to solve the case where the space component has dimension two. Stability conditions for the numerical scheme are obtained.

Exciton Multiplication from First Principles

Third-generation photovolatics require demanding cost and power conversion efficiency standards, which may be achieved through efficient exciton multiplication. Therefore, generating more than one electron-hole pair from the absorption of a single photon has vast ramifications on solar power conversion technology. Unlike their bulk counterparts, irradiated semiconductor quantum dots exhibit efficient exciton multiplication, due to confinement-enhanced Coulomb interactions and slower nonradiative losses. The exact characterization of the complicated photoexcited processes within quantum-dot photovoltaics is a work in progress. In this Account, we focus on the photophysics of nanocrystals and investigate three constituent processes of exciton multiplication, including photoexcitation, phonon-induced dephasing, and impact ionization. We quantify the role of each process in exciton multiplication through ab initio computation and analysis of many-electron wave functions. The probability of observing a multiple exciton in a photoexcited state is proportional to the magnitude of electron correlation, where correlated electrons can be simultaneously promoted across the band gap. Energies of multiple excitons are determined directly from the excited state wave functions, defining the threshold for multiple exciton generation. This threshold is strongly perturbed in the presence of surface defects, dopants, and ionization. Within a few femtoseconds following photoexcitation, the quantum state loses coherence through interactions with the vibrating atomic lattice. The phase relationship between single excitons and multiple excitons dissipates first, followed by multiple exciton fission. Single excitons are coupled to multiple excitons through Coulomb and electron-phonon interactions, and as a consequence, single excitons convert to multiple excitons and vice versa. Here, exciton multiplication depends on the initial energy and coupling magnitude and competes with electron-phonon energy relaxation. Multiple excitons are generated through impact ionization within picoseconds. The basis of exciton multiplication in quantum dots is the collective result of photoexcitation, dephasing, and nonadiabatic evolution. Each process is characterized by a distinct time-scale, and the overall multiple exciton generation dynamics is complete by about 10 ps. Without relying on semiempirical parameters, we computed quantum mechanical probabilities of multiple excitons for small model systems. Because exciton correlations and coherences are microscopic, quantum properties, results for small model systems can be extrapolated to larger, realistic quantum dots.

Wave transmission through periodic, quasiperiodic, and random one-dimensional finite lattices

The quantum mechanical transmission probability is calculated for one-dimensional finite lattices with three types of potentials: periodic, quasiperiodic, and random. When the number of lattice sites included in the computation is systematically increased, distinct features in the transmission probability vs. energy diagrams are observed for each case. The periodic lattice gives rise to allowed and forbidden transmission regions that correspond to the energy band structure of the infinitely periodic potential. In contrast, the transmission probability diagrams for both quasiperiodic and random lattices show the absence of well-defined band structures and the appearance of wave localization effects. Using the average transmissivity concept, we show the emergence of exponential (Anderson) and power-law bounded localization for the random and quasiperiodic lattices, respectively.

Avoiding Negative Probabilities in Quantum Mechanics

As currently understood since its discovery, the bare Klein-Gordon theory consists of negative quantum probabilities which are considered to be physically meaningless if not outright obsolete. Despite this annoying setback, these negative probabilities are what led the great Paul Dirac in 1928 to the esoteric discovery of the Dirac Equation. The Dirac Equation led to one of the greatest advances in our understanding of the physical world. In this reading, we ask the seemingly senseless question, "Do negative probabilities exist in quantum mechanics?" In an effort to answer this question, we arrive at the conclusion that depending on the choice one makes of the quantum probability current, one will obtain negative probabilities. We thus propose a new quantum probability current of the Klein-Gordon theory. This quantum probability current leads directly to positive definite quantum probabilities. Because these negative probabilities are in the bare Klein-Gordon theory, intrinsically a result of negative energies, the fact that we here arrive at a theory with positive probabilities, means that negative energy particles are not to be considered problematic as is the case in the bare Klein-Gordon theory. From an abstract-objective stand-point; in comparison with positive energy particles, the corollary is that negative energy particles should have equal chances to exist. As to why these negative energy particles do not exist, this is analogous to asking why is it that Dirac's antimatter does not exist in equal proportions with matter. This problem of why negative energy particles do not exist in equal proportions with positive energy particles is a problem that needs to be solved by a future theory.

(Never) Mind your p’s and q’s: Von Neumann versus Jordan on the foundations of quantum theory

In two papers entitled "On a new foundation [Neue Begr\"undung] of quantum mechanics," Pascual Jordan (1927b,g) presented his version of what came to be known as the Dirac-Jordan statistical transformation theory. As an alternative that avoids the mathematical difficulties facing the approach of Jordan and Paul A. M. Dirac (1927), John von Neumann (1927a) developed the modern Hilbert space formalism of quantum mechanics. In this paper, we focus on Jordan and von Neumann. Central to the formalisms of both are expressions for conditional probabilities of finding some value for one quantity given the value of another. Beyond that Jordan and von Neumann had very different views about the appropriate formulation of problems in quantum mechanics. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identication of sets of canonically conjugate variables, i.e., p's and q's. For von Neumann, not constrained by the analogy to classical mechanics, it required only the identication of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p's and q's. Jordan and von Neumann also stated the characteristic new rules for probabilities in quantum mechanics somewhat differently. Jordan (1927b) was the first to state those rules in full generality. Von Neumann (1927a) rephrased them and, in a subsequent paper (von Neumann, 1927b), sought to derive them from more basic considerations. In this paper we reconstruct the central arguments of these 1927 papers by Jordan and von Neumann and of a paper on Jordan's approach by Hilbert, von Neumann, and Nordheim (1928). We highlight those elements in these papers that bring out the gradual loosening of the ties between the new quantum formalism and classical mechanics.

The counterfactual process in weak values

From the viewpoint of mathematics, quantum-mechanical probability seems to be controversial. To resolve this problem, we have reconstructed the Born rule using the weak value. We also discuss its meaning.

Derivation of an amplitude of information in the setting of a new family of fractional entropies

By generalizing the basic functional equation f(xy)=f(x)+f(y) in the form fβ(xy)=fβ(x)+fβ(y), β>1, one can derive a family of solutions which are exactly the inverse of the Mittag–Leffler function, referred to as Mittag–Leffler logarithm, or logarithm of fractional order. This result provides a new family of generalized informational entropies which are indexed by a parameter clearly related to fractals, via fractional calculus, and which is quite relevant in the presence in defect of observation. The relation with Shannon’s entropy, Renyi’s entropy and Tsallis’ entropy is clarified, and it is shown that Tsallis’ generalized logarithm has a significance in terms of fractional calculus. The case β=2 looks like directly relevant to amplitude of probability in quantum mechanics, and provides an approach to the definition of “amplitude of informational entropy”. One examines the kind of result one can so obtain in applying the maximum entropy principle. In the presence of uncertain definition (or fuzzy definition) the Mittag–Leffler function would be more relevant than the Gaussian normal law. To some extent, this new formulation could be fully supported by the derivation of a new family of fractional Fisher information.

A simple model for the spin singlet: mathematical equivalence of non-locality, slave will and conspiracy

A hidden-variable model reproducing the quantum mechanical probabilities for a spin singlet is presented. The model violates only the hypothesis of independence of the distribution for the hidden variables from the detector settings and vice versa (measurement independence). It otherwise satisfies the hypotheses of setting independence, outcome independence—made in the derivation of the Bell inequality—and that of compliance with Malus's law—made in the derivation of the Leggett inequality. It is shown that the violation of the measurement independence hypothesis may be explained alternatively by assuming a non-local influence of the detector settings on the hidden variables, or by taking the hidden variable to influence the choice of settings (limitation of free will), or finally by purporting a conspiracy. It is demonstrated that the last two cases admit a realization through existing local classical resources.

The no-boundary measure in scalar–tensor gravity

In this paper, we study the no-boundary wavefunction in scalar–tensor gravity with various potentials for the non-minimally coupled scalar field. Our goal is to calculate probabilities for the scalar field—and hence the effective gravitational coupling and cosmological constant—to take specific values. Most calculations are performed in the minisuperspace approximation, and we use a saddle point approximation for the Euclidean action, which is then evaluated numerically. We find that for potentials that have several minima, none of them is substantially preferred by the quantum-mechanical probabilities. We argue that the same is true for the stable and the runaway solution in the case of a dilaton-type potential. Technically, this is due to the inclusion of quantum-mechanical effects (fuzzy instantons). These results are in contrast to the often-held view that vanishing gravitation or cosmological constants would be exponentially preferred in quantum cosmology, and they may be relevant to the cosmological constant problem and the dilaton stabilization problem.

Transport characteristics of tunnel heterostructures: Transition from the quantum to the classical limit

Two possible mechanisms for the partial or complete loss of information contained in the quantum-mechanical phase of an electron moving in a stochastic solid-state structure are examined. The first involves phase randomization of the electron characteristics (for example, by elastic scattering of electrons on defects in thin metallic layers) and the second arises from inelastic interactions of current carriers with external degrees of freedom. With a double-barrier heterostructure as an example, it is shown that in the first case the quantum-mechanical approach reduces to a semiclassical method, in which the probabilities of individual events appear, rather than the quantum-mechanical probability amplitudes. The second case corresponds to a transition to the classical theory of charge transport. The effect of decoherence on the differential conductivity and shot noise in double-barrier tunnelling systems with a superconducting electrode is evaluated and the changes in these owing to the transition from quantum to incoherent classical electron transport are analyzed.