Completeness of the Coulomb wave functions in quantum mechanics

We give an explicit, direct, and fairly elementary proof that the radial energy eigenfunctions for the hydrogen atom in quantum mechanics, bound and scattering states included, form a complete set. The proof uses only some properties of the confluent hypergeometric functions and the Cauchy residue theorem from analytic function theory; therefore it would form useful supplementary reading for a graduate course on quantum mechanics.

On the two-dimensional thermoelectric power in quantum wells of non-parabolic materials under magnetic quantization

We present a simplified theoretical formulation of the thermoelectric power (TP) under magnetic quantization in quantum wells (QWs) of nonlinear optical materials on the basis of a newly formulated magneto-dispersion law. We consider the anisotropies in the effective electron masses and the spin-orbit constants within the framework of k.p formalism by incorporating the influence of the crystal field splitting. The corresponding results for III-V materials form a special case of our generalized analysis under certain limiting conditions. The TP in QWs of Bismuth, II-VI, IV-VI and stressed materials has been studied by formulating appropriate electron magneto-dispersion laws. We also address the fact that the TP exhibits composite oscillations with a varying quantizing magnetic field in QWs of n-Cd3As2, n-CdGeAs2, n-InSb, p-CdS, stressed InSb, PbTe and Bismuth. This reflects the combined signatures of magnetic and spatial quantizations of the carriers in such structures. The TP also decreases with increasing electron statistics and under the condition of non-degeneracy, all the results as derived in this paper get transformed into the well-known classical equation of TP and thus confirming the compatibility test. We have also suggested an experimental method of determining the elastic constants in such systems with arbitrary carrier energy spectra from the known value of the TP. (C) 2010 Elsevier Ltd. All rights reserved.

A Compact Model incorporating Quantum Effects for Ultra-Thin-Body Double-Gate MOSFETs

We propose a compact model which predicts the channel charge density and the drain current which match quite closely with the numerical solution obtained from the Full-Band structure approach. We show that, with this compact model, the channel charge density can be predicted by taking the capacitance based on the physical oxide thickness, as opposed to C-eff, which needs to be taken when using the classical solution.

On the fundamentals of coherence theory

In the thesis I study various quantum coherence phenomena and create some of the foundations for a systematic coherence theory. So far, the approach to quantum coherence in science has been purely phenomenological. In my thesis I try to answer the question what quantum coherence is and how it should be approached within the framework of physics, the metatheory of physics and the terminology related to them. It is worth noticing that quantum coherence is a conserved quantity that can be exactly defined. I propose a way to define quantum coherence mathematically from the density matrix of the system. Degenerate quantum gases, i.e., Bose condensates and ultracold Fermi systems, form a good laboratory to study coherence, since their entropy is small and coherence is large, and thus they possess strong coherence phenomena. Concerning coherence phenomena in degenerate quantum gases, I concentrate in my thesis mainly on collective association from atoms to molecules, Rabi oscillations and decoherence. It appears that collective association and oscillations do not depend on the spin-statistics of particles. Moreover, I study the logical features of decoherence in closed systems via a simple spin-model. I argue that decoherence is a valid concept also in systems with a possibility to experience recoherence, i.e., Poincaré recurrences. Metatheoretically this is a remarkable result, since it justifies quantum cosmology: to study the whole universe (i.e., physical reality) purely quantum physically is meaningful and valid science, in which decoherence explains why the quantum physical universe appears to cosmologists and other scientists very classical-like. The study of the logical structure of closed systems also reveals that complex enough closed (physical) systems obey a principle that is similar to Gödel's incompleteness theorem of logic. According to the theorem it is impossible to describe completely a closed system within the system, and the inside and outside descriptions of the system can be remarkably different. Via understanding this feature it may be possible to comprehend coarse-graining better and to define uniquely the mutual entanglement of quantum systems.

Effects of Parasitics and Interface Traps on Ballistic Nanowire FET in the Ultimate Quantum Capacitance Limit

In this paper, we focus on the performance of a nanowire field-effect transistor in the ultimate quantum capacitance limit (UQCL) (where only one subband is occupied) in the presence of interface traps (D-it), parasitic capacitance (C-L), and source/drain series resistance (R-s,R-d), using a ballistic transport model and compare the performance with its classical capacitance limit (CCL) counterpart. We discuss four different aspects relevant to the present scenario, namely: 1) gate capacitance; 2) drain-current saturation; 3) subthreshold slope; and 4) scaling performance. To gain physical insights into these effects, we also develop a set of semianalytical equations. The key observations are as follows: 1) A strongly energy-quantized nanowire shows nonmonotonic multiple-peak C-V characteristics due to discrete contributions from individual subbands; 2) the ballistic drain current saturates better in the UQCL than in the CCL, both in the presence and absence of D-it and R-s,R-d; 3) the subthreshold slope does not suffer any relative degradation in the UQCL compared to the CCL, even with Dit and R-s,R-d; 4) the UQCL scaling outperforms the CCL in the ideal condition; and 5) the UQCL scaling is more immune to R-s,R-d, but the presence of D-it and C-L significantly degrades the scaling advantages in the UQCL.

Descriptions of operators in quantum mechanics

The problem of expressing a general dynamical variable in quantum mechanics as a function of a primitive set of operators is studied from several points of view. In the context of the Heisenberg commutation relation, the Weyl representation for operators and a new Fourier-Mellin representation are related to the Heisenberg group and the groupSL(2,R) respectively. The description of unitary transformations via generating functions is analysed in detail. The relation between functions and ordered functions of noncommuting operators is discussed, and results closely paralleling classical results are obtained.

Wigner distribution for angle coordinates in quantum mechanics

The method of Wigner distribution functions, and the Weyl correspondence between quantum and classical variables, are extended from the usual kind of canonically conjugate position and momentum operators to the case of an angle and angular momentum operator pair. The sense in which one has a description of quantum mechanics using classical phase‐space language is much clarified by this extension.

Paradoxical reflection in quantum mechanics

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schroedinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schroedinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau.

Quantum and non-Markovian effects in the electron transfer reaction dynamics in the Marcus inverted region

Electron transfer reactions in large molecules may often be coupled to both the polar solvent modes and the intramolecular vibrational modes of the molecule. This can give rise to a complex dynamics which may in some systems, like betaine, be controlled more by vibrational rather than by solvent effects. Additionally, a significant contribution from an ultrafast relaxation component in the solvation dynamics may enhance the complexity. To explain the wide range of behavior that has been observed experimentally, Barbara et al. recently proposed that a model of an electron transfer reaction should minimally consist of a low-frequency classical solvent mode (X), a low-frequency vibrational mode (Q), and a high-frequency quantum mode (q) (J. Phys. Chem. 1991, 96, 3728). In the present work, a theoretical study of this model is described. This study generalizes earlier work by including the biphasic solvent response and the dynamics of the low-frequency vibrational mode in the presence of a delocalized, extended reaction zone. A novel Green's function technique has been developed which allowed us to study the non-Markovian dynamics on a multidimensional surface. The contributions from the high-frequency vibrational mode and the ultrafast component in the non-Markovian solvent dynamics are found to be primarily responsible for the dramatic increase in charge transfer rate over the prediction of the classical theories that neglect both these factors. These, along with a large coupling between the reactant and the product states, may combine to render the electron transfer rate both very large and constant over a wide range of solvent relaxation rates. A study on the free energy gap dependence of the electron transfer rate reveals that the rates are sensitive to changes in the quantum frequency particularly when the free energy gap is very large.

Qualitative theorem proving in linear constraints

We know, from the classical work of Tarski on real closed fields, that elimination is, in principle, a fundamental engine for mechanized deduction. But, in practice, the high complexity of elimination algorithms has limited their use in the realization of mechanical theorem proving. We advocate qualitative theorem proving, where elimination is attractive since most processes of reasoning take place through the elimination of middle terms, and because the computational complexity of the proof is not an issue. Indeed what we need is the existence of the proof and not its mechanization. In this paper, we treat the linear case and illustrate the power of this paradigm by giving extremely simple proofs of two central theorems in the complexity and geometry of linear programming.

Quantum Random Walks without Coin Toss

We construct a quantum random walk algorithm, based on the Dirac operator instead of the Laplacian. The algorithm explores multiple evolutionary branches by superposition of states, and does not require the coin toss instruction of classical randomised algorithms. We use this algorithm to search for a marked vertex on a hypercubic lattice in arbitrary dimensions. Our numerical and analytical results match the scaling behaviour of earlier algorithms that use a coin toss instruction.

Quantum Algorithms with Fixed Points: The Case of Database Search

The standard quantum search algorithm lacks a feature, enjoyed by many classical algorithms, of having a fixed-point, i.e. a monotonic convergence towards the solution. Here we present two variations of the quantum search algorithm, which get around this limitation. The first replaces selective inversions in the algorithm by selective phase shifts of $\frac{\pi}{3}$. The second controls the selective inversion operations using two ancilla qubits, and irreversible measurement operations on the ancilla qubits drive the starting state towards the target state. Using $q$ oracle queries, these variations reduce the probability of finding a non-target state from $\epsilon$ to $\epsilon^{2q+1}$, which is asymptotically optimal. Similar ideas can lead to robust quantum algorithms, and provide conceptually new schemes for error correction.

General classical theory of spinning particles in a meson field

An exact classical theory of the motion of a point dipole in a meson field is given which takes into account the effects of the reaction of the emitted meson field. The meson field is characterized by a constant $\chi =\mu /\hslash $ of the dimensions of a reciprocal length, $\mu $ being the meson mass, and as $\chi \rightarrow $ 0 the theory of this paper goes over continuously into the theory of the preceding paper for the motion of a spinning particle in a Maxwell field. The mass of the particle and the spin angular momentum are arbitrary mechanical constants. The field contributes a small finite addition to the mass, and a negative moment of inertia about an axis perpendicular to the spin axis. A cross-section (formula (88 a)) is given for the scattering of transversely polarized neutral mesons by the rotation of the spin of the neutron or proton which should be valid up to energies of 10$^{9}$ eV. For low energies E it agrees completely with the old quantum cross-section, having a dependence on energy proportional to p$^{4}$/E$^{2}$ (p being the meson momentum). At higher energies it deviates completely from the quantum cross-section, which it supersedes by taking into account the effects of radiation reaction on the rotation of the spin. The cross-section is a maximum at E $\sim $ 3$\cdot $5$\mu $, its value at this point being 3 $\times $ 10$^{-26}$ cm.$^{2}$, after which it decreases rapidly, becoming proportional to E$^{-2}$ at high energies. Thus the quantum theory of the interaction of neutrons with mesons goes wrong for E $\gtrsim $ 3$\mu $. The scattering of longitudinally polarized mesons is due to the translational but not the rotational motion of the dipole and is at least twenty thousand times smaller. With the assumption previously made by the present author that the heavy partilesc may exist in states of any integral charge, and in particular that protons of charge 2e and - e may occur in nature, the above results can be applied to charged mesons. Thus transversely polarised mesons should undergo a very big scattering and consequent absorption at energies near 3$\cdot $5$\mu $. Hence the energy spectrum of transversely polarized mesons should fall off rapidly for energies below about 3$\mu $. Scattering plays a relatively unimportant part in the absorption of longitudinally polarized mesons, and they are therefore much more penetrating. The theory does not lead to Heisenberg explosions and multiple processes.

Emitter near an arbitrary body: Purcell effect, optical theorem and the Wheeler-Feynman absorber

The altered spontaneous emission of an emitter near an arbitrary body can be elucidated using an energy balance of the electromagnetic field. From a classical point of view it is trivial to show that the field scattered back from any body should alter the emission of the source. But it is not at all apparent that the total radiative and non-radiative decay in an arbitrary body can add to the vacuum decay rate of the emitter (i.e.) an increase of emission that is just as much as the body absorbs and radiates in all directions. This gives us an opportunity to revisit two other elegant classical ideas of the past, the optical theorem and the Wheeler-Feynman absorber theory of radiation. It also provides us alternative perspectives of Purcell effect and generalizes many of its manifestations, both enhancement and inhibition of emission. When the optical density of states of a body or a material is difficult to resolve (in a complex geometry or a highly inhomogeneous volume) such a generalization offers new directions to solutions. (c) 2012 Elsevier Ltd. All rights reserved.

Heat fluctuations and coherences in a quantum heat engine

Quantum coherence can affect the thermodynamics of small quantum systems. Coherences have been shown to affect the power generated by a quantum heat engine (QHE) which is coupled to two thermal photon reservoirs and to an additional cavity mode. We show that the fluctuations of the heat exchanged between the QHE and the reservoirs strongly depend on quantum coherence, especially when the engine operates as a refrigerator, i.e., heat current flows from the cold bath to the hot bath. Intriguingly, we find that the ratio of positive and negative (with respect to the thermodynamic force) fluctuations in the heat current satisfies a universal coherence-independent fluctuation theorem.