Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schrödinger wave function

Floyd, Edward R.

doi:10.1007/bf02190052

Cited by 39 publications

(107 citation statements)

References 12 publications

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“…We demonstrate the absence of trajectories in the derivation of the QHJE from point transformations leading to the trivial hamiltonian [11][12][13][14][15][16][17][18]. The basic point is that trajectories can only be defined by time parameterisation of them, and include the Bohm-de Broglie pilot wave representation and Floyd's time parameterisation [19][20][21][22][23][24][25][26][27] by using Jacobi theorem. We show in this paper that these time parameterisations are ill defined.…”

Section: Introductionmentioning

confidence: 93%

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Energy quantisation and time parameterisation

Faraggi

Matone

2014

Eur. Phys. J. C

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We show that if space is compact, then trajectories cannot be defined in the framework of the quantum Hamilton-Jacobi (HJ) equation. The starting point is the simple observation that when the energy is quantised it is not possible to make variations with respect to the energy, and the time parameterisation t − t 0 = ∂ E S 0 , implied by Jacobi's theorem, which leads to the group velocity, is ill defined. It should be stressed that this follows directly from the quantum HJ equation without any axiomatic assumption concerning the standard formulation of quantum mechanics. This provides a stringent connection between the quantum HJ equation and the Copenhagen interpretation. Together with tunnelling and the energy quantisation theorem for confining potentials, formulated in the framework of quantum HJ equation, it leads to the main features of the axioms of quantum mechanics from a unique geometrical principle. Similar to the case of the classical HJ equation, this fixes its quantum analog by requiring that there exist point transformations, rather than canonical ones, leading to the trivial hamiltonian. This is equivalent to a basic cocycle condition on the states. Such a cocycle condition can be implemented on compact spaces, so that continuous energy spectra are allowed only as a limiting case. Remarkably, a compact space would also imply that the Dirac and von Neumann formulations of quantum mechanics essentially coincide. We suggest that there is a definition of time parameterisation leading to trajectories in the context of the quantum HJ equation having the probabilistic interpretation of the Copenhagen School.

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Section: Introductionmentioning

confidence: 93%

“…This is the time parameterisation of particle trajectories that, as first observed by Floyd [19][20][21][22][23][24][25][26][27], should be used in considering the quantum HJ equation. This is just how trajectories are defined in CM as it implies the group velocity.…”

Section: Introductionmentioning

confidence: 98%

Energy quantisation and time parameterisation

Faraggi

Matone

2014

Eur. Phys. J. C

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“…The solution of Eq. (3), investigated also by Floyd [6,7,8,9] and Faraggi-Matone [1,2,3], is given in Ref. [5] as…”

Section: Introductionmentioning

confidence: 99%

Quantum Newton's law

Bouda¹,

Djama²

2001

Physics Letters A

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Using the quantum Hamilton-Jacobi equation within the framework of the equivalence postulate, we construct a Lagrangian of a quantum system in one dimension and derive a third order equation of motion representing a first integral of the quantum Newton's law. We then integrate this equation in the free particle case and compare our results to those of Floydian trajectories. Finally, we propose a quantum version of Jacobi's theorem.

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“…The same conclusion is also reached in ref. [1]. We would like to add that if we use the Bohm ansatz (α = 1, β = 0), eqs.…”

Section: Microstatesmentioning

confidence: 99%

“…This relation is also reproduced in ref. [8] where (1) and (2) are derived from the Schrödinger equation (SE) by appealing to the probability current. By setting α = |α| exp(ia), β = |β| exp(ib),…”

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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Where and why the generalized Hamilton-Jacobi representation describes microstates of the Schrödinger wave function

Cited by 39 publications

References 12 publications

Energy quantisation and time parameterisation

Energy quantisation and time parameterisation

Quantum Newton's law

The Three-Dimensional Quantum Hamilton-Jacobi Equation and Microstates

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