Solving Klein’s paradox

Wang, Huaiyu

doi:10.1088/2399-6528/abd340

Cited by 14 publications

(23 citation statements)

References 27 publications

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“…[10]. Third, these two equations helped us to correctly calculate the reflection coefficient of a relativistic particle with spin-0 through one-dimensional step potential [42]. The achieved reflection curves were qualitatively the same as those of a Dirac particle.…”

Section: Relativistic Motionmentioning

confidence: 77%

“…Facing the infinitely high barrier, the particle has to reflect totally. In a previous paper [42], we have solved the same potential problems for relativistic particles by means of the equations listed in Table 2 here. Now we compare the curves there with those in Figs.…”

Section: Discussionmentioning

confidence: 99%

“…/1 R m c E → . That is to say, even facing an infinitely high potential, there is still some probability for the particle to transmit, and the greater the particle's energy or the less its static mass, the greater the transmission probability, which might provide an insight why neutrinos could transmit through almost everywhere [42].…”

Section: Discussionmentioning

confidence: 99%

“…In the author's previous papers [10,42], we have pointed out that when the potential was piecewise constant, KGE could be factorized as 0 0…”

Section: Relativistic Motionmentioning

confidence: 99%

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The Modified Fundamental Equations of Quantum Mechanics in Symmetric Forms

Wang¹

2021

Preprint

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Up to now, Schrödinger equation, Klein-Gordon equation (KGE) and Dirac equation are believed the fundamental equations of quantum mechanics. Schrödinger equation has a defect that there is no NKE solutions. Dirac equation has positive kinetic energy (PKE) and negative kinetic energy (NKE) branches. Both branches should have low momentum, or nonrelativistic, approximations: one is Schrödinger equation and the other is NKE Schrödinger equation. KGE has two problems: it is an equation of second time derivative, and calculated density is not definitely positive. To overcome the problems, it should be revised as PKE and NKE decoupled KGEs. The fundamental equations of quantum mechanics after the modification have at least two merits. They are of unitary in that everyone contains the first time derivative and are symmetric with respect to PKE and NKE. This reflects the symmetry of the PKE and NKE matters, as well as matter and dark matter, of our universe. The problems of one-dimensional step potentials are resolved by means of the modified fundamental equations for a nonrelativistic particle.

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Section: Relativistic Motionmentioning

confidence: 77%

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

“…In the author's previous papers [10,42], we have pointed out that when the potential was piecewise constant, KGE could be factorized as 0 0…”

Section: Relativistic Motionmentioning

confidence: 99%

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The Modified Fundamental Equations of Quantum Mechanics in Symmetric Forms

Wang¹

2021

Preprint

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“…3. We have solved Klein's paradox [2]. The key is to piecewise solve Dirac equation in such a way that in the region where the particle's energy E is greater (less) than the potential V, the solution of the positive (negative) energy branch is adopted.…”

Section: Appendix the Negative Energy Solutions Of Dirac Equation Are Not Antiparticlesmentioning

confidence: 99%

Fundamental formalism of statistical mechanics and thermodynamics of negative kinetic energy systems

Wang¹

2021

J. Phys. Commun.

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The solutions of a particle's Dirac equation contains a negative kinetic energy (NKE) branch. Such an energy spectrum has an upper limit but no lower limit, so that the system with this spectrum, called NKE system, is of negative temperature. Fundamental formulas of statistical mechanics and thermodynamics of NKE systems are presented. All the formulas have the same forms of those of positive kinetic energy (PKE) systems. Almost all thermodynamic quantities, except entropy and specific heat, have a contrary sign compared to those of PKE systems. Specially, pressure is negative and its microscopic mechanism is given. Entropy is always positive and Boltzmann entropy formula remains valid. The three laws of thermodynamics remain valid, as long as the thermodynamic quantities have a negative sign. Negative temperature Carnot engine can work between two negative temperatures. Since the NKE levels need not be fully filled, it is argued that the concept of Dirac's Fermion Sea can be totally abandoned.

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The Dirac impenetrable barrier in the limit point of the Klein energy zone

Vincenzo

2023

J. Phys. Commun.

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We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height $V_{0}$ with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e., $E\rightarrow V_{0}-\mathrm{m}c^{2}$, for a given $V_{0}$. In such a case, the particle is actually colliding with an impenetrable barrier. In fact, $V_{0}\rightarrow E+\mathrm{m}c^{2}$, for a given relativistic energy $E\,(<V_{0})$, is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when $V_{0}\rightarrow E+\mathrm{m}c^{2}$ does not seem to be compatible with the existence of the impenetrable barrier.

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Solving Klein’s paradox

Cited by 14 publications

References 27 publications

The Modified Fundamental Equations of Quantum Mechanics in Symmetric Forms

The Modified Fundamental Equations of Quantum Mechanics in Symmetric Forms

Fundamental formalism of statistical mechanics and thermodynamics of negative kinetic energy systems

The Dirac impenetrable barrier in the limit point of the Klein energy zone

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