Modified potential and Bohm's quantum-mechanical potential

Floyd, Edward R.

doi:10.1103/physrevd.26.1339

Cited by 70 publications

(117 citation statements)

References 7 publications

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“…(6) to regain P (x r ) = N 1 x 2 r exp(−α 2 x 2 r ). In the next step, h(x r , x i ) = (x 2 r + x 2 i ) is found to be a solution of the conservation equation (7). (One could guess this expression from the form ofẋ ⋆ẋ in this case, as it appears in its denominator.)…”

Section: Examplesmentioning

confidence: 99%

“…One of the advantages of the resulting theory, which offers a new interpretation of quantum mechanics, is that it does not face the problem of stationarity of particles in bound states, encountered in the dBB representation. Another trajectory approach to quantum mechanics, which also claims the absence of this problem, is the representation developed by Floyd, Faraggi, Matone (FFM) and others [7][8][9].…”

Section: Introductionmentioning

confidence: 99%

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Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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It is shown that in the complex trajectory representation of quantum mechanics, the Born's Ψ ⋆ Ψ probability density can be obtained from the imaginary part of the velocity field of particles on the real axis. Extending this probability axiom to the complex plane, we first attempt to find a probability density by solving an appropriate conservation equation. The characteristic curves of this conservation equation are found to be the same as the complex paths of particles in the new representation. The boundary condition in this case is that the extended probability density should agree with the quantum probability rule along the real line. For the simple, time-independent, one-dimensional problems worked out here, we find that a conserved probability density can be derived from the velocity field of particles, except in regions where the trajectories were previously suspected to be nonviable. An alternative method to find this probability density in terms of a trajectory integral, which is easier to implement on a computer and useful for single particle solutions, is also presented. Most importantly, we show, by using the complex extension of Schrodinger equation, that the desired conservation equation can be derived from this definition of probability density.

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Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Probability and complex quantum trajectories

John¹

2009

Annals of Physics

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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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“…Floyd remarked this problem and proposed to use a trigonometric representation in the real wave function cases [8,9]. He also proposed that quantum trajectories were obtained by using Jacobi's theorem [9,10],…”

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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Modified potential and Bohm's quantum-mechanical potential

Cited by 70 publications

References 7 publications

Probability and complex quantum trajectories

Probability and complex quantum trajectories

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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