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arXiv:1603.07726 · European Journal of Physics

Revisiting double Dirac delta potential

Zafar Ahmed, Sachin Kumar, Mayank Sharma, Vibhu Sharma

Published Jun 10, 2016• 20 citations
arXivHTMLPDF

Abstract

Abstract We study a general double Dirac delta potential to show that this is the simplest yet still versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ( T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of a Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission (or zero reflectivity, R ( E ) ) at energy E = 0 is non-intuitive. However, this has been found earlier and called the ‘threshold anomaly’. Here we show that it is a critical phenomenon and we can have 0 ≤ R ( 0 ) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke a zero-energy and zero curvature eigenstate ( ψ ( x ) = Ax + B ) of the delta well between two symmetric rigid walls for R ( 0 ) = 0 . We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.

DOI
10.1088/0143-0807/37/4/045406

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Abstract We study a general double Dirac delta potential to show that this is the simplest yet still versatile solvable potential to introduce double wells, avoided crossings, resonances and perfect transmission ( T = 1). Perfect transmission energies turn out to be the critical property of symmetric and anti-symmetric cases wherein these discrete energies are found to correspond to the eigenvalues of a Dirac delta potential placed symmetrically between two rigid walls. For well(s) or barrier(s), perfect transmission (or zero reflectivity, R ( E ) ) at energy E = 0 is non-intuitive. However, this has been found earlier and called the ‘threshold anomaly’. Here we show that it is a critical phenomenon and we can have 0 ≤ R ( 0 ) < 1 when the parameters of the double delta potential satisfy an interesting condition. We also invoke a zero-energy and zero curvature eigenstate ( ψ ( x ) = Ax + B ) of the delta well between two symmetric rigid walls for R ( 0 ) = 0 . We resolve that the resonant energies and the perfect transmission energies are different and they arise differently.
We consider the energy eigenvalue problem of the Dirac delta well potential, V ( x ) = − V 0 δ( x ), placed between two rigid walls at x = ± a . When the strength parameter is increased, the ground state eigenvalue, E 0 , passes from a positive to a negative value. We show that when s = 1, E 0 = 0, the eigenstate is forced to be of zero-curvature: ψ 0 ( x ) = A ( a − | x |), and it is orthogonal to all excited states. On the other hand, if V 0 is made negative then a zero-energy and zero-curvature eigenstate does not exist.
The solution of the one-dimensional Dirac equation is examined for the symmetrical double delta-function potential. A formula for the relativistic transmission coefficient is derived from the solution. The maximum condition as well as the minimum condition for the transmission coefficient is determined. The transmission coefficient is shown to be equal to unity in its maximum. In its minimum, it is smaller than the transmission coefficient for the single delta-function potential of the twofold strength. Further, some differences between the relativistic and non-relativistic transmission coefficient are presented. It is found that the wave numbers, at which the maxima and minima occur according to the non-relativistic treatment, are shifted by the relativistic treatment. Unlike the non-relativistic transmission coefficient, the relativistic transmission coefficient does not approach unity in the range of high energies. It oscillates between unity and a constant. The value of this constant depends on the strength of the delta-interaction. Thus, the relativistic corrections to the transmission coefficient for the symmetrical double delta-function potential are found to represent significant shifts in its characteristics.
We revisit a rectangular barrier as well as a rectangular well (pit) between two rigid walls. The former is the well known double-well potential and the latter is a hole potential. Let ∣V0∣|V_0|∣V0​∣ be the height (depth) of the barrier (well) then for a fixed geometry of the potential, we show that in the double-well, E=V0(>0)E=V_0(>0)E=V0​(>0), and in the hole potential (V0<0V_0 <0V0​<0), E=0E=0E=0, can be energy eigenvalues provided V0V_0V0​ admits some special discrete values. These states have been missed out earlier which emerge only when one seeks the special zero-energy solution of one-dimensional Schr{ö}dinger equation as ψ(x)=Bx+Cψ(x)=Bx+Cψ(x)=Bx+C.
In one-dimensional scattering, the portion of particles that is transmitted in general vanishes as the kinetic energy of the incident particles approaches zero. Based upon published results from the literature, the present work shows that for a potential consisting of two Dirac δ functions of arbitrary strength, a finite portion of the incident particles is transmitted at threshold for certain choices of the set of parameters defining the potential. The underlying reasons are shown to be related to the structure of the bound levels of the potential. It is shown that for a potential with a finite range but of general shape the reflection coefficient in general tends to unity near threshold, which is what is usually observed. However, in the presence of a bound level at the onset of the continuum, a finite portion of the incident particles is transmitted at threshold for a potential with a general shape and, furthermore, it can be shown that if the potential V is symmetric with V(x)=V(−x) then resonance transmission is observed, i.e., the reflection coefficient vanishes at threshold.
Regarding the quantum mechanical transmission–reflection problem in one dimension with a potential of the form of the derivative of the Dirac delta function δ'(x) = dδ(x)/dx, Christiansen et al recently found that, depending on how δ'(x) is interpreted, there can be a resonance which leads to partial transmission. This is in contrast to the earlier consensus that such a potential allows no transmission. The δ'(x) can be regarded as the narrow-width limit of a certain function Δ'(x) of a finite range. Christiansen et al assumed a rectangular function for Δ'(x). We examine various other forms and how the resonance depends on the shape of Δ'(x). We also present some general observations related to the 'threshold anomaly'.
Dirac particles can undergo perfect transmission through a sufficiently high potential barrier in the Klein zone. Although the perfect Klein tunneling (often referred to as the Klein paradox) is similar to the non-relativistic resonant transmission which occurs only when the kinetic energy exceeds the barrier, the underlying mechanism is believed to be fundamentally distinct. In this work, we show that for the relativistic double-barrier model the perfect-transmission curve can pass continuously from the above-barrier zone to the Klein zone. Additionally, in the Klein zone, perfect transmission occurs even for subcritical barrier heights, supported by both bound-state analysis and wave-packet dynamics. These findings suggest a connection between perfect Klein tunneling and resonant transmission, and provide new insights into the physical nature of the Klein paradox.