WKB Wave Research Articles

Quantum mechanical metastability: When and why?

We study quantum mechanical metastability with an eye towards false vacuum decay. We point out some technical and conceptual problems with the familiar bounce treatment of this process. We illustrate with simple quantum mechanical examples that the bounce formalism fails to account for the correct boundary conditions. It is also shown, that the bounce approach overestimates the time scales for tunneling of localized packets in typical (slightly) biased double well potentials. We present a thorough WKB analysis with particular attention to semiclassical trajectories corresponding to complex saddle points. We point out that the boundary conditions determine the proper choice of saddle points and the bounce approach fails to account for semiclassical trajectories in many physically relevant cases. We recognize that these saddle points account for the matching conditions of the WKB wave functions beyond the barriers and restore unitarity and reality of eigenvalues for self-adjoint boundary conditions. We provide a novel approach to the semiclassical analysis of out of equilibrium decay in real time in quantum statistical mechanics.

Wigner function and decoherence in quantum cosmology.

We investigate the retrieval of classical behavior from the wave function of the Universe. It has been previously argued that to do this two conditions are necessary: decoherence (or lack of quantum interference) and the existence of classical correlations. Using the Wigner function we study the mutual compatibility of these requirements. Assuming the correlation interpretation of Geroch and Hartle, we show that some form of coarse graining is essential for the theory to predict classical behavior from WKB wave functions. We also show that there must be a compromise between the degree of decoherence and the sharpness of classical correlations. A minisuperspace model is used as an example to illustrate these points.

Strong-coupling analog of the Born series in terms of a modified WKB approach.

In this paper we incorporate different strengths of the centrifugal potential into a single partial wave function of the WKB approach in different spatial regions. By a suitable choice of these regions and strengths, the WKB wave functions can be made bounded and smooth everywhere as well as compatible with quantum mechanics both close to and far from the origin. This modified WKB approximation becomes exact in the strong-coupling limit. Moreover, by iteration, it develops an absolute convergent series expansion of the correct scattering wave function at fixed coupling constants.

Wigner's function and tunneling

We construct Wigner's functions of a system with the potential— 1 2 ω 2q 2 to exhibit and resolve some paradoxical features present in this description. Among others, we show that tunneling at negative energies arises through real trajectories associated with positive energies, in contradistinction of the usual WKB picture where this effect comes about by using complex trajectories of the correct energy. We resolve this puzzle by showing (a) that the initial data of this quantal description already contain “wrong” energy regions, (b) the interference phenomenon of WKB wave functions which provides the main contribution to Wigner's function in the semiclassical domain becomes strongly non-local in the presence of a separatrix, which not only allows, but necessitates the presence of these “wrong” classical trajectories in the present description.

WKB theory of multiphoton processes in atoms

The WKB theory of multiphoton processes in atoms is formulated by using classical perturbation theory. We discuss the approximations used here and give a short review of the results of this theory in the case of a monochromatic field. The present WKB theory has a wider range of validity than the usual quantum perturbation one. The one-dimensional WKB wave function for an arbitrary time-dependent electromagnetic field is obtained.

Wentzel-Kramers-Brillouin method in multidimensional tunneling.

The WKB method is commonly used in semiclassical approximations to the wave function in both the classically allowed and the forbidden regions of a one-dimensional potential. In a multidimensional space, the method can be adapted to construct wave functions in an ``allowed'' region from classical trajectories or wave normals. However, in the ``forbidden'' region the WKB wave function is in general specified by two sets of wave fronts, the equiphase and the equiamplitude surfaces or equivalently by two sets of paths defined to be normal to these surfaces, respectively. We present a Huygens-type construction for obtaining these wave fronts and paths, which reveals that for non-normal incidence the paths are coupled to each other. The analysis enables us to answer some of the basic questions concerning tunneling in multidimensional nonseparable potentials. A special and important case occurs when the incident wave is normal to the turning surface. We show that for normal incidence the path equations are decoupled and are equivalent to Newton's equations of motion for the inverted potential and energy.

A MULTI-DIMENSIONAL TUNNELING THEORY WITH APPLICATION TO SCANNING TUNNELING MICROSCOPE

We have developed a formal general approach for evaluating the WKB wave function in a forbidden region of a non-separable potential in a three-dimensional space. The wave function is described by two sets of orthogonal wave-fronts, the equi-phase and equiamplitude surfaces, or equivalently by two sets of paths defined to be normal to these surfaces respectively. We have extended a Huygens type construction to the forbidden region to obtain the multi-dimensional wave function. The construction determines the wave-fronts and the paths. It is found that the equations for the paths obtained from the construction are coupled to each other and do not reduce to a set of ordinary differential equations. However, if the incident wave is normal to the turning surface, the equations satisfied by the paths can be de-coupled. These equations are found to be equivalent to Newton's equations of motion for the inverted potential and energy. This characteristics has been used to calculate the current density distribution for an STM. In this paper, we use the same approach to derive the expressions for the lateral resolutions and corrugations for a model STM.

Computation of quantum mechanical transmission probability from plane interfaces using electronic band structures

A method for computing the quantum mechanical transmission probability (QMTP) from plane interfaces is presented. The method uses the electronic band structure of the material and the independent-electron approximation, and consists of a proper reduction of the three-dimensional problem to a one-dimensional problem, using a generalized form of the WKB wave functions, and matching the wave functions at the interface to preserve the continuity of a generalized probability current and density. The method was used to compute the QMTP at the interface of a simple metal and the Si(100) and Si(111) surface for ballistic electrons traveling normal to the interface at the Schottky image barrier maximum. The result shows that the crystal orientation and the location of the band minimum affect the QMTP.

A new class of analytically solvable quantum scattering problems

Suppose the energyk2 and the potentialV(r) specify in partial wavel a QM scattering problem that cannot be solved in closed form. The relevant WKB wave functions, in turn, exactly solve for the givenk2 andl a radial Schroedinger equation. The potentialWl(k, r) involved in it can be made well-behaved by suitable cut-off. The set of nonintegrable cut-off scattering problems can thus be mapped on a set of integrable ones.

Correlations in the wave function of the Universe.

The Everett-type interpretations of quantum mechanics and quantum cosmology proposed independently by Hartle, Geroch, and Wada are discussed. They essentially involve regarding a strong peak in the wave function as a definite prediction. Wave functions in quantum cosmology are usually peaked about correlations between coordinates and momenta, and methods for identifying such correlations are introduced. The first method involves Wigner's function, a quantum-mechanical analogue of the classical phase-space distribution. The properties of this distribution are discussed and it is shown how it can be of use in describing the emergence of classical behavior from quantum systems. The second method involves a suitably chosen canonical transformation. These methods are applied to harmonic-oscillator examples, which are of relevance to scalar field fluctuations in inflationary universe models. These methods are also applied to WKB wave functions in quantum mechanics and quantum cosmology. The manner in which the wave function becomes peaked about sets of classical solutions is elucidated. This is extended to include inhomogeneous perturbations about minisuperspace in quantum cosmology, and the derivation of the semiclassical Einstein equations, ${G}_{\ensuremath{\mu}\ensuremath{\nu}}$=8\ensuremath{\pi}G〈${T}_{\ensuremath{\mu}\ensuremath{\nu}}$〉, from the Wheeler-DeWitt equation is considered. A condition under which they are valid is derived. It is essentially the requirement that the distribution of ${T}_{\ensuremath{\mu}\ensuremath{\nu}}$, as a function of the matter modes, is strongly peaked about its average value. Some situations in which this condition is satisfied are discussed.

A note on traversal time for tunneling

A simple approach to study the traversal time for tunneling is given. By using the WKB wave function to evaluate the velocity field of particles in the barrier region, an expression for the traversal time τ=edx[m/2(V(x)-E)]1/2 is obtained in conformity with the recent results.

A rotational-dependent analytical solution to the dissociative state: Application to b 3Σ+u state of H2

The rotational-dependent potential for a dissociative state is represented by U(r)=U0+B1/r +B2/r2+[N(N+1)−Λ2]/2Mr2. An analytical solution ψE(r) of the Schrödinger radial equation, valid for all regions of internuclear distance r and energy E, is obtained in terms of confluent hypergeometric function of the complex arguments. The solution is evaluated by expanding the confluent hypergeometric function onto a basis set of shifted Chebyshev polynomials. The expansion coefficients are recovered by a backward recursion technique. The summation process of Chebyshev polynomials converts a slowly convergent series or a divergent asymptotic series into a rapidly convergent one. The solution thus obtained is applied to calculate the vibrational wave function of the dissociative b 3Σ+u state of H2 to compare with the previous semiclassical WKB wave function. The solution of the rotational-corrected Morse potential is used for the upper bound c 3Πu state. The bound-continuum Frank–Condon overlap amplitude is computed as a function of energy E for different rotational quantum numbers N. Its dependence on N is found to be significant for large N. The decay rate of the metastable c 3Π+u (v=0), via perturbative mixing with b 3Σ+u, computed here with exact wave functions, is an order of magnitude smaller than the previous semiclassical value. However, the decay rate via forbidden radiative transitions to b 3Σ+u is close to the previous value. Radiative transition to b 3Σ+u is now believed to be the predominant decay mode of the metastable c 3Π+u state (at v=0). Lifetimes of the fine structure levels of N=1 and N=2 obtained are 1.00 ms for J=N and 1.31–1.32 ms for J=N±1. The lifetimes of the predissociative c 3Π−u (v=0) state are 2.33×10−8 s for N=1 and 7.65×10−9 s for N=2.

Quantum effects in the trajectories of sputtered ions.

Recent experimental results of Yu have shown remarkable behavior of the sputtering yield as a function of ion velocity. The usual assumption that the sputtered ions desorb as classical particles with a constant velocity is clearly incorrect at low velocities. We have tried a simple model based on a conjecture that the ion follows a quantum rather than a classical trajectory. The results, which are based on quasiclassical and quasistationary final-state ion trajectories (WKB wave functions with a complex potential), are in excellent agreement with the experiment. However, the imaginary part of the complex force necessary to achieve this agreement is at least an order of magnitude larger than typical interatomic forces. The conclusion therefore is that these quantum effects on the ion trajectory are not the cause of the nonlinearity of Yu's data.

Symplectically invariant WKB wave functions.

Traditional WKB theory yields wave functions which have divergences and are not invariant under any reasonable class of transformations. This Letter presents alternative WKB wave functions which have no divergences and are symplectic invariants.

Anharmonic oscillator: A study of the perturbation series for the wave function.

The perturbation series for the ground-state wave function of the anharmonic oscillator has been reexamined. Rigorous analysis enables the factoring of behavior dominant in the true eigenfunction. In particular, the perturbation problem is found to reduce significantly if the asymptotic WKB wave function of zero order is employed as the zero-order wave function.

A Semi-Classical Treatment of the Coupled Channel Equation with Non-Local Interaction. I

Coupled channel problem in one spatial dimension with non·local potentials of general type is treated by the WKB method. This is done by generalizing the WKB treatment of the single channel problem with non-local potentials proposed before and applied with great success to the interactions of the resonating . group method. Properties of the obtained WKB wave function are studied in detail and the conservation of the flux is investigated. Characters and roles of two kinds of turning points are studied in detail by comparing with the coupled channel problem with local potentials.

A new uniform semiclassical wave function for single surface and multisurface scattering

A procedure for the evaluation of one-dimensional uniform semiclassical wave functions is explored. This form of the approximate wave function is tested for the case of an exponential potential and is compared to the exact quantum mechanical and WKB wave functions. It is found that the uniform approximation provides a very accurate representation of the exact wave function, even at the turning points of the classical motion, where the WKB approximation diverges. It is demonstrated how correction terms can be constructed, leading to a formal expansion for the exact wave function. The individual terms of this expansion each involve a finite number of reflection points with uniform semiclassical propagation between these reflections. The analysis is extended to multisurface (nonadiabatic) problems, removing the turning point singularities which are generally present in semiclassical formulations of nonadiabatic problems.

A uniform WKB approximation for spheroidal wave functions

Uniform WKB wave functions are developed for prolate spheroidal wave functions which arise when the Schrödinger equation is given in prolate spheroidal coordinates. A detailed numerical test is presented for angular spheroidal wave functions. It shows that the approximate uniform WKB wave functions are excellent in accuracy.

An Improved Model for Post-Collision Interaction (PCI) and High Resolution Ar LMM Auger Spectra Revealing New PCI Effects

An improved semi-classical model for post-collision interaction (PCI) in photo-excited Auger decay is presented. A simple analytical expression is obtained by using complex WKB wave functions for the slow electron.The model gives the right asymtotic line-shape at high excess energies and the energy shift of the peak maximum is in agreement with experimental results. New high resolution argon LMM Auger spectra excited by electrons at different excess energies (4.5-2751.0 eV) relative the L3 level in Ar are presented. Extra structures are observed at low excess energies and are tentatively interpreted to be caused by "shake-down" processes of the slow electron. The relative shifts between Kr M5N2,3N2,3(1S 0) and Ar L3M2,3M2,3(1D 2) lines are measured and tabulated. The full widths and the asymmetry parameters at half and quarter maximum are also given.

Path integrals and the WKB approximation

The WKB approximation is demonstrated to be formally identical to the semiclassical limit of the Feynman path integral propagator. In the limit that the path integral propagator is dominated by the classical particle trajectory, it is identical to the propagator calculated directly from WKB wave functions in the stationary phase approximation.