Trajectories in the Context of the Quantum Newton's Law

Bouda, A.; Djama, T.

doi:10.1238/physica.regular.066a00097

Cited by 12 publications

(47 citation statements)

References 18 publications

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“…Now, if we take the orthogonal projection P 0 of P on the relativistic trajectory (a = 1, b = 0) and compute the distance P P 0 we get Since Eq. (23) indicates thatẋ is a monotonous function, then, whatever the point P we have |t P − t P 0 | < t n+1 − t n for every RQT [10]. So, at the classical limit (h → 0), because t n+1 − t n → 0, then |t P − t P 0 | → 0 and P P 0 → 0.…”

Section: Motion Under the Constant Potentialmentioning

confidence: 96%

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Relativistic quantum motion

Djama¹

2006

Phys. Scr.

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Using the relativistic quantum stationary Hamilton-Jacobi equation within the framework of the equivalence postulate, and grounding oneself on both relativistic and quantum Lagrangians, we construct a Lagrangian of a relativistic quantum system in one dimension and derive a third order equation of motion representing a first integral of the relativistic quantum Newton's law. Then, we plot the relativistic quantum trajectories of a particle moving under the constant and the linear potentials. We establish the existence of nodes and link them to the de Broglie's wavelength.

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Section: Motion Under the Constant Potentialmentioning

confidence: 96%

“…It is a generalization of the one exposed in Refs. [9,10,11]. So, we have derived the fundamental relativistic quantum Newton's law expressed into Eqs.…”

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confidence: 99%

Relativistic quantum motion

Djama¹

2006

Phys. Scr.

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“…On the other hand, many suggestions to formulate the quantum trajectory equations were proposed [3,4,15,17,18,19,20]. In a recent paper [21], the QSHJE in one dimension, 1 2m…”

Section: Introductionmentioning

confidence: 99%

The Three-Dimensional Quantum Hamilton-Jacobi Equation and Microstates

Bouda

Meziane

2006

Int J Theor Phys

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In a stationary case and for any potential, we solve the three-dimensional quantum Hamilton-Jacobi equation in terms of the solutions of the corresponding Schrödinger equation. Then, in the case of separated variables, by requiring that the conjugate momentum be invariant under any linear transformation of the solutions of the Schrödinger equation used in the reduced action, we clearly identify the integration constants successively in one, two and three dimensions. In each of these cases, we analytically establish that the quantum Hamilton-Jacobi equation describes microstates not detected by the Schrödinger equation in the real wave function case.

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“…Furthermore, in Ref. [12], it is shown that the time delay in tunneling through a potential barrier, calculated by using Jacobi's theorem as proposed by Floyd, manifests also some ambiguities.…”

Section: Introductionmentioning

confidence: 99%

“…This means that when the particle gets to these points, it will stand still forever. Another feature which seems to be unsatisfactory is the extreme rapid divergence of the velocity in the classically forbidden regions which is manifested for the three different potentials considered in [12]. Other comments about relation (6) are given by Floyd in [15].…”

Section: Introductionmentioning

confidence: 99%

From a Mechanical Lagrangian to the Schrödinger Equation: A Modified Version of the Quantum Newton Law

Bouda

2003

Int. J. Mod. Phys. A

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In the one-dimensional stationary case, we construct a mechanical Lagrangian describing the quantum motion of a non-relativistic spinless system. This Lagrangian is written as a difference between a function T , which represents the quantum generalization of the kinetic energy and which depends on the coordinate x and the temporal derivatives of x up the third order, and the classical potential V (x). The Hamiltonian is then constructed and the corresponding canonical equations are deduced. The function T is first assumed arbitrary. The development of T in a power series together with the dimensional analysis allow us to fix univocally the series coefficients by requiring that the well-known quantum stationary Hamilton-Jacobi equation be reproduced. As a consequence of this approach, we formulate the law of the quantum motion representing a new version of the quantum Newton's law. We also analytically establish the famous Bohm's relation µẋ = ∂S0/∂x outside of the framework of the hydrodynamical approach and show that the well-known quantum potential, although it is a part of the kinetic term, it plays really a role of an additional potential as assumed by Bohm.

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Trajectories in the Context of the Quantum Newton's Law

Cited by 12 publications

References 18 publications

Relativistic quantum motion

Relativistic quantum motion

The Three-Dimensional Quantum Hamilton-Jacobi Equation and Microstates

From a Mechanical Lagrangian to the Schrödinger Equation: A Modified Version of the Quantum Newton Law

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