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Journal of Physics A Mathematical and General

Barrier penetration via perturbation theory

Asher Peres

Published Sep 1, 1980• 2 citations
Publisher

Abstract

It is shown that Fermi's golden rule is valid for any initial state having a very narrow energy spread even if it is not an eigenstate of the free Hamiltonian. As an example, the author computes explicitly the tunnelling rate through a rectangular barrier. This method may have application in the Stark effect, autoionisation, fission, and similar problems.

DOI
10.1088/0305-4470/13/9/024

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It is shown that Fermi's golden rule is valid for any initial state having a very narrow energy spread even if it is not an eigenstate of the free Hamiltonian. As an example, the author computes explicitly the tunnelling rate through a rectangular barrier. This method may have application in the Stark effect, autoionisation, fission, and similar problems.
A simple way to derive Bardeen’s tunneling theory is introduced. With a low rate of tunneling of, for example, electrons from one electrode to another through a wide vacuum gap, the appearance of the second electrode can be regarded as a perturbation. Then Bardeen’s formula for tunneling follows exactly from Fermi’s golden rule, supposing that this perturbation appears suddenly at some instant of time. Bardeen’s theory is illustrated by assuming a smoothly variable barrier shape, including the rectangular one as a limiting case. This exactly solvable example clarifies the sensitivity of the tunneling probability to the barrier shape and the energy of the tunneling particle.
The rates of charge transfer and energy transfer are essential for understanding nanoscale phenomena and processes. Here we introduce the general conditions for the emergence of rate processes in quantum mechanics. We refer to the generic scenario in which a transition is induced between eigenstates of a given Hamiltonian by a weak perturbation. Analysis reveals that when the initial and final states are pure states, the transition rate is time-dependent and fails to reach a finite constant value. If, however, the final state is a mixed ensemble that is sufficiently wide and dense in energy, a rate constant emerges, given by Fermi’s Golden Rule (FGR). When the initial state is also mixed, for example, a thermal equilibrium state, a thermal rate constant is formulated in terms of transition rates between specific eigenstates, summed over the final eigenstates, and averaging over the thermal distribution of the initial eigenstates.
Often one needs to calculate the evolution time of a state under a Hamiltonian with no explicit time dependence when only numerical methods are available. In cases such as this, the usual application of Fermi's golden rule and first-order perturbation theory is inadequate as well as being computationally inconvenient. Instead, what one needs are conditions under which the evolution time may be obtained from the easily calculated energy uncertainty. This work derives some general conditions for obtaining the evolution time from the energy uncertainty.
In scanning tunneling microscopy (STM), the potential barriers are often close to or lower than the Fermi level, where both semiclassical tunneling theory and Bardeen’s tunneling theory are no longer valid. We present a new perturbation approach for general quantum transmission phenomena. For thick, positive potential barriers, it reduces to Bardeen’s tunneling theory. For potential barriers lower than the energy level, it is valid as long as the barrier thickness is smaller than one half of the de Broglie wavelength, a condition which is always satisfied in STM. An immediate application is shown.
A simple analysis of the bound-free system shows that for a narrow continuum the Fermi golden rule can be generalised for the high-intensity regime. This generalisation has the form of a simple expression for the decay rate and helps to explain nearly perfect population trapping in the case of a narrowly peaked matrix element. Such population trapping means that the decay rate of a bound state strongly coupled to a continuum tends to zero for increasing coupling strength only if the matrix element falls off faster than the Lorentzian. The authors also find new solutions for pulsed excitation giving exact population trapping in the bound-free transition.