Tunneling Wave Function Research Articles

Exponentially accurate semiclassical tunneling wavefunctions in one dimension

We study the time behavior of wavefunctions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading-order component of the wavefunction that tunnels. It is exponentially small in 1/ℏ. For a wide variety of incoming wave packets, the leading-order tunneling component is Gaussian for sufficiently small ℏ. We prove this for both the large time asymptotics and for moderately large values of the time variable.

Electron tunneling in respiratory complex I

NADH:ubiquinone oxidoreductase (complex I) plays a central role in the respiratory electron transport chain by coupling the transfer of electrons from NADH to ubiquinone to the creation of the proton gradient across the membrane necessary for ATP synthesis. Here the atomistic details of electronic wiring of all Fe/S clusters in complex I are revealed by using the tunneling current theory and computer simulations; both density functional theory and semiempirical electronic structure methods were used to examine antiferromagnetically coupled spin states and corresponding tunneling wave functions. Distinct electron tunneling pathways between neighboring Fe/S clusters are identified; the pathways primarily consist of two cysteine ligands and one additional key residue. Internal water between protein subunits is identified as an essential mediator enhancing the overall electron transfer rate by almost three orders of magnitude to achieve a physiologically significant value. The identified key residues are further characterized by sensitivity of electron transfer rates to their mutations, examined in simulations, and their conservation among complex I homologues. The unusual electronic structure properties of Fe(4)S(4) clusters in complex I explain their remarkable efficiency of electron transfer.

Rectifying Effect in Polar Conjugated Molecular Junctions: A First-Principles Study

The electronic transport properties of polar conjugated molecules with different molecular lengths sandwiched between two gold electrodes are investigated using a fully self-consistent non-equilibrium Green's function method combined with density functional theory. The calculated current-voltage curve shows obvious asymmetric feature with the increasing of the molecular length. The observed asymmetric characteristic results from the dissimilar shift of the perturbed molecular energy levels and the spatial asymmetry of tunneling wave functions under external bias voltage. The rectifying effect can be further enhanced by introducing an asymmetric coupling.

Microscopic study of impurity resonance and tunneling magnetoresistance of nanoscale junctions

In this paper, we present a microscopic study of impurity resonance and tunneling magnetoresistance (TMR) for nanojunctions. We employ the Green's-function method to derive the tunneling wave functions and TMR from the spintronic model with nonmagnetic impurities and at a finite bias voltage. The analytical expressions for both direct and impurity-resonance tunnelings are obtained, where the lateral effect has been included. Within this framework, the TMR can be determined directly by the basic quantities of the junction, e.g., the position and the energy level of the impurity, the height and width of the barrier, the Fermi level and polarization of the electrodes, as well as the applied voltage. If the cross sectional area of the junction gets very small, we find that it is the impurity resonance and the quantum-coherence effect that control the bias dependence of TMR: the impurity resonance makes the TMR change from positive to negative, and the quantum-coherence effect decreases the TMR as the bias voltage increases. The experimental results of TMR for the junctions with the area smaller than $0.01\phantom{\rule{0.3em}{0ex}}\ensuremath{\mu}{\mathrm{m}}^{2}$ can then be explained naturally. We also find that the direct and impurity-resonance tunnelings will compete with each other when the area of the junction is enlarged. Taking into account both the contributions of direct tunneling and impurity-resonance tunneling, the total TMR agrees quite well with the more recent experiments on the larger-area nanojunctions.

Rectification Mechanism in Diblock Oligomer Molecular Diodes

We investigated a mechanism of rectification in diblock oligomer diode molecules that have recently been synthesized and showed a pronounced asymmetry in the measured I-V spectrum. The observed rectification effect is due to the resonant nature of electron transfer in the system and the localization properties of bound state wave functions of resonant states of the tunneling electron interacting with an asymmetric molecule in an electric field. The asymmetry of the tunneling wave function is enhanced or weakened depending on the polarity of the applied bias. The conceptually new theoretical approach, the Green's function theory of sub-barrier scattering, is able to provide a physically transparent explanation of this rectification effect based on the concept of the bound state spectrum of a tunneling electron. The theory predicts the characteristic features of the I-V spectrum in qualitative agreement with experiment.

Quantum Birth of the Rotating Universe

Quantum birth of the Universe of the Bianchi type IX filled by a rotating anisotropic liquid is studied. The Wheeler–De Witt equation is derived for the model under consideration. The tunneling wave function as a solution to the equation is found using the WKB method. The tunneling coefficient of the Universe is calculated. The probabilities of quantum birth of the Universe with and without rotation are compared for different formulations of the problem.

Complex-Domain Semiclassical Theory and Periodically Perturbed Barrier Tunneling Problems

In recent papers we demonstrated a new class of barrier tunneling phenomena which can be observed in periodically driven one dimensional barrier systems (i.e., 1.5D barrier tunneling systems). In such systems the tunneling component is in general accompanied by remarkable interference fringe patterns as the perturbation strength exceeds a certain characteristic value. Such a phenomenon is quite general and is not restricted to periodically driven 1D systems, and it may be also observed in intrinsic two dimensional autonomous systems. The complex-domain semiclassical theory developed by ourselves was applied to the problem, and we succeeded in reconstructing every detail of the fringed tunneling wavefunction only by using the complexified classical tunneling trajectories. This fact greatly motivates us to investigate the mechanism of the tunneling in terms of “classical mechanics”. The later part of the present paper is devoted to asserting the generality of the “classical mechanism” of the fringed tunneli...

WKB WAVE FUNCTIONS WITH THE INDUCED GRAVITY THEORY

The Wheeler–DeWitt equation for the induced gravity theory is constructed in the mini superspace approximation, and then solved using the WKB method under three types of boundary condition proposed respectively by Hartle & Hawking ("no boundary"), Linde and Vilenkin ("tunneling from nothing"). It is found that no matter how the gravitational and cosmological "constants" vary in the classical models, they will acquire constant values when the universe comes from quantum creation, and that, in particular, the resulting tunneling wave function under the Linde or Vilenkin boundary condition reaches its maximum value if the cosmological constant vanishes.

Wave Functions for Arbitrary Operator Ordering in the de Sitter Minisuperspace Approximation

We derive exact series solutions for the Wheeler-DeWitt equation corresponding to a spatially closed Friedmann-Robertson-Walker universe with cosmological constant for arbitrary operator ordering of the scale factor of the universe. The resulting wave functions are those relevant to the approximation which has been widely used in two-dimensional minisuperspace models with an inflationary scalar field for the purpose of predicting the period of inflation which results from competing boundary condition proposals for the wave function of the universe. The problem that Vilenkin’s tunneling wave function is not normalizable for general operator orderings, is shown to persist for other values of the spatial curvature, and when additional matter degrees of freedom such as radiation are included.

Operator ordering and consistency of the wave function of the Universe

We demonstrate in the context of the minisuperspace model consisting of a closed Friedmann-Robertson-Walker universe coupled to a scalar field that Vilenkin's tunneling wave function can only be consistently defined for particular choices of operator ordering in the Wheeler-DeWitt equation. The requirement of regularity of the wave function has the particular consequence that the probability amplitude, which has been used previously in the literature in discussions of issues such as the prediction of inflation, is likewise ill defined for certain choices of operator ordering with Vilenkin's boundary condition. By contrast, the Hartle-Hawking no-boundary wave function can be consistently defined within these models, independently of operator ordering. The significance of this result is discussed within the context of the debate about the predictions of semiclassical quantum cosmology. In particular, it is argued that inflation cannot be confidently regarded as a ``prediction'' of the tunneling wave function, for reasons similar to those previously invoked in the case of the no-boundary wave function. A synthesis of the no-boundary and tunneling approaches is argued for.

Quantum creation of an open inflationary universe

We discuss a dramatic difference between the description of the quantum creation of an open universe using the Hartle-Hawking wave function and the tunneling wave function. Recently Hawking and Turok have found that the Hartle-Hawking wave function leads to a universe with Omega = 0.01, which is much smaller that the observed value of Omega > 0.3. Galaxies in such a universe would be about $10^{10^8}$ light years away from each other, so the universe would be practically structureless. We will argue that the Hartle-Hawking wave function does not describe the probability of the universe creation. If one uses the tunneling wave function for the description of creation of the universe, then in most inflationary models the universe should have Omega = 1, which agrees with the standard expectation that inflation makes the universe flat. The same result can be obtained in the theory of a self-reproducing inflationary universe, independently of the issue of initial conditions. However, there exist two classes of models where Omega may take any value, from Omega > 1 to Omega << 1.

Bremsstrahlung inαDecay

We present the first fully quantum mechanical calculation of photon radiation accompanying charged particle decay from a barrier resonance. The soft-photon limit agrees with the classical results, but differences appear at next-to-leading-order. Under the conditions of alpha-decay of heavy nuclei, the main contribution to the photon emission stems from Coulomb acceleration and may be computed analytically. We find only a small contribution from the tunneling wave function under the barrier.

In defense of the “tunneling” wave function of the universe

The tunneling approach to the wave function of the universe has been recently criticized by Bousso and Hawking who claim that it predicts a catastrophic instability of de Sitter space with respect to pair production of black holes. We show that this claim is unfounded. First, we argue that different horizon size regions in de Sitter space cannot be treated as independently created, as they contend. And second, the WKB tunneling wave function is not simply the `inverse' of the Hartle-Hawking one, except in very special cases. Applied to the related problem of pair production of massive particles, we argue that the tunneling wave function leads to a small constant production rate, and not to a catastrophy as Bousso and Hawking's argument would suggest.

Multidimensional tunneling in terms of complex classical mechanics: Wave functions, energy splittings, and decay rates in nonintegrable systems

A semiclassical theory of multidimensional tunneling is formulated to calculate the tunneling wave function, energy splitting in a double well and decay rate constant from a metastable state. First, the tunneling wave function is calculated by analytic continuation of a quantized torus prepared in analytic forms using either the Chapman–Garrett–Miller method or the Birkhoff–Gustavson normal form method. For a weakly nonintegrable system, tunnelings are confirmed to be classified into two qualitatively different domains; pure tunneling in the I region and mixed tunneling in the C region. Semiclassical wave functions agree with quantum mechanical ones within a few percent both in classically allowed and tunneling regions. Breakdown of this simple picture is exemplified for cases of relatively strong couplings. Second, expressions of the tunneling energy splitting in a double well potential and the decay rate from a metastable state are derived. The wave function near a well formulated above is connected with the semiclassical Green’s function in the deep tunneling region. The latter is expressed by complex trajectories which start from the complex quantized torus. A preliminary numerical comparison with the quantum mechanical value is also done for the energy splitting.

Tunneling boundary condition and topology change.

In this paper the tunneling boundary condition is reexamined in terms of the Morse function. The use of this function permits one, on the one hand, to describe the shape of the singularity when topology change is allowed, and, on the other hand, to define, in the same situation, a slicing procedure which allows one to define the boundary of superspace so that the tunneling wave function of the Universe which results is defined unambiguously. \textcopyright{} 1995 The American Physical Society.

Quantum Kaluza-Klein cosmology in the Vilenkin theory

In this paper, we discuss the quantum Kaluza-Klein cosmological model in the Vilenkin theory, and give the corresponding tunnelling wave function.

Tunneling splittings in model 2D potentials. III. . Generalization to N-dimensional case

Abstract Vibrational tunneling spectra for a ‘linear’ potential are calculated with the WKB trajectory method for a wide range of coupling strengths between coordinates, and compared to the quantum mechanical computations. Reasonable agreement is found between the WKB and quantum results, both in the tunneling splittings and wave functions at the ‘path-generated’ dividing line. The method is generalized to N-dimensional symmetric potentials with two equivalent minima. The results are exemplified by calculating the vibrational tunneling spectrum in a 3D ‘squeezed’ potential. The inhomogeneous broadening of tunneling doublets in supersonic beams is discussed. In rotationally excited states, the coupling strength depends on the rotational quantum number, because of the vibration-rotation interaction. The resultant line width is formed by an ensemble of coherently tunneling systems with a distribution of coupling strengths determined by the finite rotational temperature.

Self-adjoint Wheeler-DeWitt operators, the problem of time, and the wave function of the Universe.

We discuss minisuperspace aspects a non empty Robertson-Walker universe containing scalar matter field. The requirement that the Wheeler-DeWitt (WDW) operator be self adjoint is a key ingredient in constructing the physical Hilbert space and has non-trivial cosmological implications since it is related with the problem of time in quantum cosmology. Namely, if time is parametrized by matter fields we find two types of domains for the self adjoint WDW operator: a non trivial domain is comprised of zero current (Hartle-Hawking type) wave functions and is parametrized by two new parameters, whereas the domain of a self adjoint WDW operator acting on tunneling (Vilenkin type) wave functions is a {\em single} ray. On the other hand, if time is parametrized by the scale factor both types of wave functions give rise to non trivial domains for the self adjoint WDW operators, and no new parameters appear in them.

The nuclear quadrupole coupling constants and the structure of the para–para ammonia dimer

Expressions are derived for the nuclear quadrupole splittings in the E3 and E4 (para–para) states of (NH3)2 and it is shown that these can be matched with the standard expressions for rigid rotors with two identical quadrupolar nuclei. The matching is exact only when the off-diagonal Coriolis coupling is neglected. However, the selection rules for rotational transitions are just opposite to those for the rigid rotor. Hyperfine splittings are measured for the J=2←1 transitions in the E3 and E4 states with ‖K‖=1; the quadrupole coupling constants χaa=0.1509(83) MHz and χbb−χcc=2.8365(83) MHz are extracted from these measurements by the use of the above mentioned correspondence with the rigid rotor expressions. The corresponding results are also calculated, with and without the Coriolis coupling, from the six-dimensional vibration–rotation–tunneling (VRT) wave functions of (NH3)2, which were previously obtained by Olthof et al. [E.H.T. Olthof, A. van der Avoird, and P.E.S. Wormer, J. Chem. Phys. 101, 8430 (1994)]. From the comparison of χaa with the measured value it follows that the semiempirical potential and the resulting VRT states of Olthof et al. are very accurate along the interchange (ϑA,ϑB) coordinate. From χbb−χcc it follows that this potential is probably too soft in the dihedral angle γ̄=γA−γB, which causes the torsional amplitude to be larger than derived from the experiment.

Quantum State during and after Nucleation of an O(4)-Symmetric Bubble

We investigate the quantum state of a scalar field interacting with another field whose false vacuum decays through a Euclidean O(4)-symmetric bubble. We apply the field theoretical formulation of the multi-dimensional WKB tunneling wave function to the system and construct appropriate mode functions which describe the quantum state during and after the false vacuum decay. We find that the quantum state inside the nucleated bubble is Lorentz-invariant but very different from the Minkowski vacuum, containing field excitations which behave like thermal radiation at late times. Implications of the result are discussed; in particular, in connection with the Lorentz invariance of the resulting state, the possibility of creating an open universe through nucleation of an O(4)-symmetric bubble is mentioned.