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

Coutinho, Francisco Antônio Bezerra; Nogami, Y.; Toyama, F. M.

doi:10.1590/s1806-11172009000400004

Cited by 18 publications

(10 citation statements)

References 13 publications

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“…where the top and bottom components of the column vector Ψ, ϕ and χ, are given in terms of ψ in Eq. (13).…”

Section: The Mean Value Off In Thementioning

confidence: 99%

On the mean value of the force operator for 1D particles in the step potential

Vincenzo

2021

Rev. Bras. Ensino Fís.

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In the one-dimensional Klein-Fock-Gordon theory, the probability density is a discontinuous function at the point where the step potential is discontinuous. Thus, the mean value of the external classical force operator cannot be calculated from the corresponding formula of the mean value. To resolve this issue, we obtain this quantity directly from the Klein-Fock-Gordon equation in Hamiltonian form, or the Feshbach-Villars wave equation. Not without surprise, the result obtained is not proportional to the average of the discontinuity of the probability density but to the size of the discontinuity. In contrast, in the one-dimensional Schrödinger and Dirac theories this quantity is proportional to the value that the respective probability density takes at the point where the step potential is discontinuous. We examine these issues in detail in this paper. The presentation is suitable for the advanced undergraduate level.

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“…where the top and bottom components of the column vector Ψ, ϕ and χ, are given in terms of ψ in Eq. (13).…”

Section: The Mean Value Off In Thementioning

confidence: 99%

On the mean value of the force operator for 1D particles in the step potential

Vincenzo

2021

Rev. Bras. Ensino Fís.

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“…The implications of such situations regarding the transmission-reflection problem in one-dimensional quantum mechanics are reviewed at length in Ref. [69]. In the limit that l → 0 the hyperbolic secant potential smoothly approaches a δ-function potential, thus making it an ideal approximation.…”

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

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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.

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“…As a matter of fact, v has a wider meaning than δ in the sense that [Coutinho et al, 2009, Eq. ( 7 n) is the nth derivative of f with respect to x ∈ at a given point).…”

Section: Preliminaries Throughout We Definementioning

confidence: 99%