On time derivatives for &lt;X^&gt; and &lt;p^&gt;: formal 1D calculations

Vincenzo, Salvatore De

doi:10.1590/s1806-11172013000200008

Cited by 9 publications

(15 citation statements)

References 11 publications

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“…We have seen various studies on this subject in Schrödinger's theory (see, for example, Refs. [14][15][16][17] and references therein), and in Dirac's theory [5]. However, in the Klein-Fock-Gordon theory, as far as we know, there are not many studies that involve the mean value of this operator.…”

Section: Resultsmentioning

confidence: 99%

“…On the other hand, in Ref. [14], the result in Eq. (A1) was obtained directly from the Ehrenfest theorem for a (free) Schrödinger particle on a half line (in our case, the region x ∈ (−∞, 0]).…”

Section: Appendix Amentioning

confidence: 92%

“…Thus, from Eqs. (16) and (17), and using the continuity of ψ and ψ x at x = 0, we obtain the following matricial boundary conditions:…”

Section: The Mean Value Off In Thementioning

confidence: 99%

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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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Section: Resultsmentioning

confidence: 99%

Section: Appendix Amentioning

confidence: 92%

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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 non-relativistic (or Schrödinger) Ehrenfest theorem (in one dimension) was recently treated in Ref. [6] in a similar manner to that used in the present article.…”

Section: Introductionmentioning

confidence: 99%

Operators and bilinear densities in the Dirac formal 1D Ehrenfest theorem

Vincenzo¹

2015

J. Phys. Stud.

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We calculate formal time derivatives of the mean values of the standard position, velocity, and mechanical momentum operators, i. e., the Ehrenfest theorem for a one-dimensional Dirac particle in the coordinate representation. We show that these derivatives contain boundary terms that essentially depend on the values taken there by characteristic bilinear densities. We do not automatically take the boundary terms to vanish (as is usually done); nevertheless, we relate the boundary terms to similar terms that must be zero if one requires the hermiticity of certain specific unbounded operators. Throughout the article, we thoroughly discuss and illustrate all these aspects, which include the relations to certain boundary conditions. To clarify, we call our approach formal because all operations involving operators (for example, some operators products) are performed without respecting the restrictions imposed by the sets of functions on which the self-adjoint operators can act. Moreover, the Dirac Hamiltonian that we consider in our calculations contains a potential that is the time component of a Lorentz two-vector; nevertheless, we also obtain and concisely discuss the Ehrenfest theorem for a Hamiltonian with the most general Lorentz potential in (1+1) dimensions.

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“…The most common example is the potential energy of the simple harmonic oscillator [7]. In other cases, such as the infinite well and infinite step potentials, verification is problematic [8][9][10][11]. Although the Ehrenfest theorem provides a (formal) general relationship between classical and quantum dynamics, it does not necessarily (neither sufficiently) characterize the classical regime [12].…”

Section: Introductionmentioning

confidence: 99%

Classical path from quantum motion for a particle in a transparent box

Vincenzo

2014

Rev. Bras. Ensino Fís.

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We consider the problem of a free particle inside a one-dimensional box with transparent walls (or equivalently, along a circle with a constant speed) and discuss the classical and quantum descriptions of the problem. After calculating the mean value of the position operator in a time-dependent normalized complex general state and the Fourier series of the function position, we explicitly prove that these two quantities are in accordance by (essentially) imposing the approximation of high principal quantum numbers on the mean value. The presentation is accessible to advanced undergraduate students with a knowledge of the basic ideas of quantum mechanics. Keywords: correspondence principle, classical limit, Ehrenfest theorem.Consideramos o problema de uma partícula livre no interior de uma caixa unidimensional com paredes transparentes (ou equivalentemente, ao longo de um círculo com uma velocidade constante) e discutimos as descrições clássica e quântica do problema. Depois de calcular o valor médio do operador da posição num estado geral complexo normalizado dependente do tempo e a série de Fourier da função de posição, provamos explicitamente que estas duas quantidades estão em correspondência se (essencialmente) impusermos sobre o valor médio a aproximação dos números quânticos principais elevados. A apresentaçãoé acessível a alunos de graduação avançados com conhecimento das idéias básicas da mecânica quântica. Palavras-chave: princípio da correspondência, limite clássico, teorema de Ehrenfest.

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On time derivatives for <X^> and <p^>: formal 1D calculations

Cited by 9 publications

References 11 publications

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

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

Operators and bilinear densities in the Dirac formal 1D Ehrenfest theorem

Classical path from quantum motion for a particle in a transparent box

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