C.R. Leavens

Transmission, reflection and dwell times within Bohm's causal interpretation of quantum mechanics

Abstract Although many approaches have been developed within the orthodox interpretation of quantum mechanics for calculating the mean times τ T and τ R spent by transmitted and reflected particles inside a one-dimensional barrier, a completely satisfactory solution remains elusive. In this paper it is argued that Bohms causal or trajectory interpretation provides a simple, well-defined, and unambiguous method for calculating transmission, reflection and dwell times with physically reasonable properties.

Traversal times for rectangular barriers within Bohm's causal interpretation of quantum mechanics

Abstract In sharp contrast to the conventional interpretation of quantum mechanics, Bohms causal or trajectory interpretation provides a unique, well-defined, and unambiguous prescription for calculating the average time spent by a transmitted particle inside a one-dimensional barrier. Numerical results for such traversal times are presented for the special case of a particle described initially by a minimum-uncertainty-product wavepacket incident on a rectangular barrier. These are very different from the corresponding Larmor clock or time-modulated barrier results unless the transition probability is close to unity.

Extension to arbitrary barriers of the Büttiker-Landauer characteristic barrier interaction times

Abstract In a series of papers Buttiker and Landauer used various approaches to derive characteristic times describing the interaction of particles with a rectangular barrier (more general barriers were treated only within the WKB approximation). We generalize their results to arbitrary barriers and present calculated characteristic times for a selection of frequently used barrier shapes. We find that, even for a rectangular barrier, two of the approaches lead to corresponding characteristic times that can be quite different, particularly at low energy. Moreover, the identification of one of the characteristic times with the well known “dwell” time is shown not to hold in general, although it is certainly correct for a rectangular barrier.

Calculations of the superconducting isotope effect

Abstract The strong-coupling theory of superconductivity is used, without simplification, to compute the isotope effect exponent β in T c ∝ M −β for several metals. A comparison is made with the values obtained using two approximate equations for T c .

First principles calculations of the low temperature electrical resistivity of potassium

Abstract The results of first principles calculations of the low temperature (2 K ⩽ T ⩽20 K) phonon-limited electrical resistivity of potassium are presented for both the Bloch limit (possible phonon drag effects are completely ignored) and the phonon drag limit (phonon-phonon collisions, which tend to suppress phonon drag, are neglected). In the former case the agreement with experiment is very good; in the latter, the calculated results are much too low. These results are at serious variance with the conclusions of Kaveh and Wiser and reopen the question of whether or not there is any evidence for important phonon drag effects in the low temperature electrical resistivity of potassium.

Vacuum tunnelling thermopower: normal metal electrodes

Abstract Theoretical results are presented for the vacuum tunnelling “thermopower” S  limit ΔT→O V (Δ T ) Δ T where V(ΔT) is the voltage required to null the thermal tunneling current generated by maintaining a temperature difference ΔT between the electrodes of the tunnel junction. To a good approximation the tunnelling thermopower depends on temperature T and vacuum gap d over a large range of these parameters only in the combination Td. This could, at least in principle, provide the basis of a direct method for measuring absolute vacuum gaps in scanning tunnelling microscopy.

A local version of the Larmor clock

Abstract Buttikers analysis of the Larmor clock is extended to the local situation in which a uniform transverse magnetic field is confined to an arbitrary part of a barrier. The resulting local characteristic electron-barrier interaction times for transmission and reflection are complex quantities whose real and imaginary parts are measurable, at least in principle. Analytic expressions for the local interaction times of a rectangular barrier are derived and their properties investigated. The effects of a multiple image potential are examined numerically. Quite apart from the question of local interaction times, it is shown that in the tunnelling regime the transverse magnetic field produces significant precession only when it extends to within a decay length κ−1 of either edge of a rectangular barrier.

A least upper bound on the superconducting transition temperature

Abstract It is rigorously shown that the superconducting transition temperature of any material for which the Eliashberg theory is valid must satisfy kBTc ⩽ 0.2309 A, where A is the area under its electron-phonon spectral function α2F(ω). This relation is a least upper bound, not just an upper bound, in the sense that there is an optimal situation in which the equality holds. This occurs when the Coulomb pseudopotential parameter μ ∗ is zero and the spectral function is the Einstein spectrum Aδ(ω − 1.750 A). These results are generalized in an approximate, but sufficiently accurate, way to the case μ ∗ ≠ 0 to obtain the more useful least upper bound k B T c ⩽ c(μ ∗ ) A and the corresponding optimal spectrum Aδ[ω − d(μ ∗ )A] . Numerical results for the functions c(μ ∗ ) and d(μ∗) are presented for 0 ⩽ μ ∗ ⩽ 0.20 . It is shown that the Tcs of many materials (including Nb3Sn), for which experimental values of A and μ ∗ are available, do not lie very far below the upper bound.

The time-modulated barrier approach to traversal times from the Bohm trajectory point of view

Abstract Buttiker and Landauer introduced the time-modulated barrier method for determining the average traversal time τ T for an electron incident on a static barrier. This approach was motivated by the expectation that with the addition to the barrier of a small oscillatory component, V 1 cosωt, the tunneling behaviour would exhibit distinct low (ω⪡τ T −1 ) and high (ω⪢τ T −1 ) frequency regimes. Recently De Raedt, Garcia and Huyghebaert studied the scattering of gaussian wave packets by time-modulated rectangular barriers. They were unable to extract τ T from the frequency dependence of the calculated transmission probability |T(ω)| 2 . In the present paper, possible reasons are suggested. In particular, it is argued that, at least within the Bohm trajectory interpretation of quantum mechanics, a very small increase in the transmission probability actually involves atypically long reflection times rather than the average transmission time of the unperturbed barrier. Hence, there is no reason to expect a signature of τ T in |T(ω)| 2 . The Bohm trajectory approach is then used to investigated the ω dependence of the average traversal time and the distribution of traversal times both of which are found to exhibit interesting resonant behavior.

Application of the quantum clock of salecker and Wigner to the tunneling time problem

Abstract Peres used Salecker and Wigners prescription for a quantum clock in a theoretical study of time-of-flight determination of the velocity of a free particle. In the present paper, the quantum clock is applied to a closely related problem, determination of the average time spent inside a one-dimensional potential barrier V(z)Θ(z)Θ(d-z) by initially free “incident” electrons of energy E. For an opaque rectangular barrier straightforward application of the approach leads to a “clocked” result that differs by orders of magnitude from the result postulated by Buttiker for the average “intrinsic” dwell time τD(0,d;E). It is shown that this difference can be eliminated by appropriate choice of initial state for the ensemble of identical clocks and by applying to their average behaviour when coupled to tunneling particles the calibration determined for the corresponding ensemble of freely running clocks. The difference that persists for more transparent barriers is attributed, following Peres, to perturbation of the clock and/or particle dynamics during the measurement process. It is most serious in the limit d→0 where the ratio of clocked to intrinsic times peaks at a value of about 1.6. Because of the nonlinear relation between “actual” time (i.e. the parameter t in the Schrodinger equation) and uncalibrated clock time it does not seem possible to decompose the mean dwell time for an opaque barrier as “measured” by the quantum clock approach considered here into individual components associated with transmitted and reflected particles. This is consistent with the point of view that the “tunneling time”, which refers only to the transmitted particles, is not a meaningful concept within conventional interpretations of quantum mechanics.