Proper treatment of the delta function potential in the one-dimensional Dirac equation

Calkin, M. G.; Kiang, D.; Nogami, Y.

doi:10.1119/1.15031

Cited by 44 publications

(40 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(3) does not hold for the wave function [2][3][4]. This is essentially for the same reason as the one found in GW's example with Eq.…”

Section: The Dirac Delta Functionsupporting

confidence: 61%

“…(3) may turn out to be inconsistent with the definition of f (x) itself. Such an unusual situation was recognized in relation to the Dirac equation in one dimension [2][3][4]. More recently Griffiths and Walborn (GW) illustrated such a situation by means of a simple mathematical example [5].…”

Section: The Dirac Delta Functionmentioning

confidence: 99%

“…It has also been known that if the potential part of the equation is of the form of δ(x)ψ(x), where ψ(x) is the Dirac wave function, and if it is replaced by δ(x) ψ(y)δ(y)dy, then Eq. (3) can be used [2,3,6]. The interaction in this form can be interpreted as a zero-range "separable potential".…”

Section: The Dirac Delta Functionmentioning

confidence: 99%

“…Equation (3) implies that, although δ(x) itself is not well-defined for x = 0, when it occurs as a factor in an integrand, the integral has a well-defined value.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Unusual situations that arise with the Dirac delta function and its derivative

Coutinho

Nogami

Toyama

2009

Rev. Bras. Ensino Fís.

View full text Add to dashboard Cite

There is a situation such that, when a function ƒ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">) is combined with the Dirac delta function δ(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), the usual formula <img src="/img/revistas/rbef/v31n4/a04form01.gif" align="absmiddle">does not hold. A similar situation may also be encountered with the derivative of the delta function δ'(<img src="/img/revistas/rbef/v31n4/a04x.gif" align="absmiddle">), regarding the validity of <img src="/img/revistas/rbef/v31n4/a04form02.gif" align="absmiddle">. We present an overview of such unusual situations and elucidate their underlying mechanisms. We discuss implications of the situations regarding the transmission-reflection problem of one-dimensional quantum mechanics.

show abstract

“…(3) does not hold for the wave function [2][3][4]. This is essentially for the same reason as the one found in GW's example with Eq.…”

Section: The Dirac Delta Functionsupporting

confidence: 61%

Section: The Dirac Delta Functionmentioning

confidence: 99%

Section: The Dirac Delta Functionmentioning

confidence: 99%

“…Equation (3) implies that, although δ(x) itself is not well-defined for x = 0, when it occurs as a factor in an integrand, the integral has a well-defined value.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Unusual situations that arise with the Dirac delta function and its derivative

Coutinho

Nogami

Toyama

2009

Rev. Bras. Ensino Fís.

View full text Add to dashboard Cite

show abstract

“…In certain instances, the one-dimensional Dirac equation, which has a wave function defined by a differential equation involving the delta function, can result in the usual definition of the delta function being inconsistent with the definition of the wave function itself [66][67][68]. The implications of such situations regarding the transmission-reflection problem in one-dimensional quantum mechanics are reviewed at length in Ref.…”

Section: Bound Modes In a Model Potentialmentioning

confidence: 99%

Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential

Hartmann

Portnoi

2014

Phys. Rev. A

View full text Add to dashboard Cite

We analyze bound modes of two-dimensional massless Dirac fermions confined within a hyperbolic secant potential, which provides a good fit for potential profiles of existing top-gated graphene structures. We show that bound states of both positive and negative energies exist in the energy spectrum and that there is a threshold value of the characteristic potential strength for which the first mode appears. Analytical solutions are presented in several limited cases and supercriticality is discussed.

show abstract

On Relativistic Transmissions through a Symmetrical Pair of Delta-Barriers or Delta-Wells

Yanetka

2002

phys. stat. sol. (b)

View full text Add to dashboard Cite

73.40.Gk The solution of the one-dimensional Dirac equation is examined for the symmetrical double deltafunction 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.

show abstract

Proper treatment of the delta function potential in the one-dimensional Dirac equation

Cited by 44 publications

References 0 publications

Unusual situations that arise with the Dirac delta function and its derivative

Unusual situations that arise with the Dirac delta function and its derivative

Quasi-exact solution to the Dirac equation for the hyperbolic-secant potential

On Relativistic Transmissions through a Symmetrical Pair of Delta-Barriers or Delta-Wells

Contact Info

Product

Resources

About