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Mathematical Proceedings of the Cambridge Philosophical Society

Phase-integral treatment of transmission through an inverted Morse potential

Bengt Lundborg

Published May 1, 1977• 20 citations
Publisher

Abstract

Abstract The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.

DOI
10.1017/s0305004100053536

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The transmission coefficient for a one-dimensional system of N potential barriers of general shapes (unequal or equal) is expressed exactly, by means of the phase-integral method of N. Fröman and P. O. Fröman, in terms of quantities characterizing the separate barriers and wells of the system. The exact formula, being of interest for further rigorous studies of transmission properties, can for evaluation readily be converted into a useful approximate formula by insertion of available phase-integral expressions (of an arbitrary order) for the characteristic quantities mentioned above. Error estimates are given. The sub- and superbarrier cases are treated in a unified way. Some key facts about the phase-integral method are given in Appendix A.

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This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.

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A real-time functional-integral method is used to derive an effective action that gives the transmission spectrum of a tunneling particle interacting with a bath of harmonic oscillators. The transmission spectum is expressed in terms of double functional integrals with respect to the coordinate of the particle which are evaluated by means of the stationary-phase approximation. The equations of motion for the stationary-phase trajectories are solved exactly for an arbitrary spectral density function of the bath, and the obtained solutions are used to find the transmission spectra for specific examples. For a bath with single frequency \ensuremath{\omega}, an analytic expression of the transmission spectrum is obtained which covers from the sudden tunneling (\ensuremath{\omega}${\mathit{T}}_{0}$\ensuremath{\ll}1) to the adiabatic one (\ensuremath{\omega}${\mathit{T}}_{0}$\ensuremath{\gg}1), where ${\mathit{T}}_{0}$ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with an Ohmic spectrum, the differential tunneling conductance at low bias voltage V and for \ensuremath{\eta}${\mathit{T}}_{0}$\ensuremath{\ll}1 is found to obey a power law \ensuremath{\sim}(${\mathit{eVT}}_{0}$/\ensuremath{\Elzxh}${)}^{\mathrm{\ensuremath{\eta}}{\mathit{T}}_{0}{\mathit{S}}_{0}/2\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{\Elzxh}}}$, where \ensuremath{\eta} is the friction coefficient and ${\mathit{S}}_{0}$ is the tunneling exponent in the absence of interaction. \textcopyright{} 1996 The American Physical Society.

arXiv:cond-mat/9607189•Physical review. B, Condensed matter•Sep 15, 1996•10 citations
Abstract The transmission of a quantal particle through a barrier obtained by inverting a Morse potential is studied by means of the ‘symmetric’ phase-integral approximations introduced by N. Fröman and further developed by N. Fröman and P. O. Fröman. The calculations are performed up to the fifth-order approximation. Approximate expressions for important quantities such as transmission and reflexion coefficients, phases of the wave function and also for quantities related to the phenomenon of resonances, which may occur when the wave function is given as a standing wave, are compared to the corresponding exact quantities, and this comparison shows the great accuracy of these phase-integral approximations.
The transmission coefficient for a one-dimensional system of N potential barriers of general shapes (unequal or equal) is expressed exactly, by means of the phase-integral method of N. Fröman and P. O. Fröman, in terms of quantities characterizing the separate barriers and wells of the system. The exact formula, being of interest for further rigorous studies of transmission properties, can for evaluation readily be converted into a useful approximate formula by insertion of available phase-integral expressions (of an arbitrary order) for the characteristic quantities mentioned above. Error estimates are given. The sub- and superbarrier cases are treated in a unified way. Some key facts about the phase-integral method are given in Appendix A.
This work is a contribution to the theory of the quantum tunneling effect. In the literature at least two barriers are studied, for which the third-order contribution to the phase - integral asymptotic approximation of the transmission coefficient do not contribute. These are the parabolic barrier and the inverse Morse barrier. In the present work we will show that with a proper choice of the so called base function there is at least one more barrier in this category namely the Eckart-Epstein potential. The fact that the third - order contribution vanishes is a good indication that we have found an optimal choice of the base function, and the treatment to find an optimal base function may be possible to generalize to other classes of potential barriers. For particles of a low energy compared to the energy near the top of the barrier we obtain a vary low transmission coefficient, which means that the probability for tunneling to occur is very low. There exist some cases, for example that with a double barrier which is transparent, even for certain relatively low energies but no evidence for this kind of transparency for a single barrier has been found. The present work does not give any such evidence. At the same time there are still speculations on cold fusion like effects, which would demand a higher probability for tunneling through for a single barrier.
Quantum above-barrier reflection of ultra-cold atoms by the Rosen-Morse potential is analytically considered within the mean field Gross-Pitaevskii approximation. Reformulating the problem of reflectionless transmission as a quasi-linear eigenvalue problem for the potential depth, an approximation for the specific height of the potential that supports reflectionless transmission of the incoming matter wave is derived via modification of the Rayleigh-Schroedinger time-independent perturbation theory. The approximation provides highly accurate description of the resonance position for all the resonance orders if the nonlinearity parameter is small compared with the incoming particles chemical potential. Notably, the result for the first transmission resonance turns out to be exact, i.e., the derived formula for the resonant potential height gives the exact value of the first nonlinear resonances position for all the allowed variation range of the involved parameters, the nonlinearity parameter and chemical potential. This has been shown by constructing the exact solution of the problem for the first resonance. Furthermore, the presented approximation reveals that, in contrast to the linear case, in the nonlinear case reflectionless transmission may occur not only for potential wells but also for potential barriers with positive potential height. It also shows that the nonlinear shift of the resonance position from the position of the corresponding linear resonance is approximately described as a linear function of the resonance order. Finally, a compact (yet, highly accurate) analytic formula for the n-th order resonance position is constructed via combination of analytical and numerical methods.
Transmission and reflection of a quantal particle by a single-hump potential barrier are analysed by means of an amplitude-phase decomposition of the wavefunction on both sides of the barrier. The amplitude-phase analysis of the wavefunction provides a particular invariant of the Ermakov–Lewis type, which originates in the matching process. The transmission and reflection coefficients turn out to be simple functions of this invariant. Numerical calculations of the invariant for an Eckart–Epstein potential barrier provide very accurate results.
Abstract The simplest representation for the quantum mechanical wavefunction in terms of classical quantities is due to Jeffreys (1925), Wentzel (1926), Kramers (1926), and Brillouin (1926a,b). It generalizes the free-motion form, ψ = exp(ipx/h), by allowing for variation in the momentum p(x) as a function of position, a correction that replaces the exponent by a phase integral, ∫p(x) dx, and introduces a normalization term related to the local velocity. A similar description may also be applied in classically forbidden regions by allowing the momentum to become imaginary, p(x) = i|p(x)|, provided that the local de Broglie wavelength, λ.(x) = h/p(x), varies sufficiently slowly with x. Details are given in Section 2.1 and discussed at greater length by Froman and Froman (1967) and Froman (1980).
A real-time functional-integral method is used to derive an effective action that gives the transmission spectrum of a tunneling particle interacting with a bath of harmonic oscillators. The transmission spectum is expressed in terms of double functional integrals with respect to the coordinate of the particle which are evaluated by means of the stationary-phase approximation. The equations of motion for the stationary-phase trajectories are solved exactly for an arbitrary spectral density function of the bath, and the obtained solutions are used to find the transmission spectra for specific examples. For a bath with single frequency \ensuremath{\omega}, an analytic expression of the transmission spectrum is obtained which covers from the sudden tunneling (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1) to the adiabatic one (\ensuremath{\omega}T0{\mathit{T}}_{0}T0​\ensuremath{\gg}1), where T0{\mathit{T}}_{0}T0​ is the time it would take a classical particle to traverse the inverted bare potential barrier. For a bath with an Ohmic spectrum, the differential tunneling conductance at low bias voltage V and for \ensuremath{\eta}T0{\mathit{T}}_{0}T0​\ensuremath{\ll}1 is found to obey a power law \ensuremath{\sim}(eVT0{\mathit{eVT}}_{0}eVT0​/\ensuremath{\Elzxh}${)}^{\mathrm{\ensuremath{\eta}}{\mathit{T}}_{0}{\mathit{S}}_{0}/2\mathrm{\ensuremath{\pi}}\mathrm{\ensuremath{\Elzxh}}}$, where \ensuremath{\eta} is the friction coefficient and S0{\mathit{S}}_{0}S0​ is the tunneling exponent in the absence of interaction. \textcopyright{} 1996 The American Physical Society.