Time-independent, paraxial and time-dependent Madelung trajectories near zeros

Berry, Michael

doi:10.1088/1751-8121/ad10f2

Cited by 4 publications

(4 citation statements)

References 38 publications

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“…Therefore, an interesting approach could be to apply the quantum potential approach to study the properties of waves with locally stable caustics of the other types. Third, in a recent work, Berry [26] has remarked that the patterns of Bohm trajectories can be completely different when the same wave equation describes different physical contexts. That is, for the timeindependent Schrödinger or Helmholtz equation the trajectories circulate around the phase singularities of the wave function; in the paraxially approximate wave, they spiral slowly in or out of the zeros as well as circulating.…”

Section: Discussionmentioning

confidence: 99%

Classical trajectories from the zeros of the quantum potential: the 2D isotropic harmonic oscillator

Silva-Ortigoza,

Ortiz-Flores,

Teresa Sosa-Sánchez

et al. 2024

Phys. Scr.

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In the first part of this work, using the quantum potential approach, we show that a solution to the time-independent Schr"odinger equation determines a subset of classical solutions, only if \textit{the region corresponding to the zeroes of the quantum potential is tangent to the caustic region determined by the classical trajectories}. Thus, the solutions of the time-independent Schr"odinger equation, according to their caustic and the zeros of the quantum potential, can be classified in three different cases given by the following conditions: the two regions coincide, they are tangent at certain subset of points, and the two regions are not tangent at any point. In the second part, as examples of the first type of wave functions, we present the solutions of the Schr"odinger equation for the $2D$ isotropic harmonic oscillator, which are eigenfunctions of both the Hamiltonian and the angular momentum operators. That is, we show that for this family of solutions, the zeroes of the quantum potential coincide with the caustic region. Furthermore, we find that the classical trajectories, determined from the quantum ones and the zeroes of the quantum potential, conform to a family of elliptical curves for a particle with energy, $(2n + l +1) \hbar \omega$, and orbital angular momentum $l \hbar$.

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Section: Discussionmentioning

confidence: 99%

Classical trajectories from the zeros of the quantum potential: the 2D isotropic harmonic oscillator

Silva-Ortigoza,

Ortiz-Flores,

Teresa Sosa-Sánchez

et al. 2024

Phys. Scr.

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“…The patterns considered here do not exhaust the possibilities. For approximations to solutions of the time-independent wave equations, streamlines near phase singularities can form spirals rather than closed loops [31], and Madelung streamlines for time-dependent waves avoid each phase singularity in spacetime, except for one that meets it in a cusp [31,32].…”

Section: Discussionmentioning

confidence: 99%

Kinetically anisotropic Hamiltonians: plane waves, Madelung streamlines and superpositions

Berry

2024

Eur. J. Phys.

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A Hamiltonian in two space dimensions whose kinetic-energy contributions have opposite signs is studied in detail. Solutions of the time-independent Schrödinger equation for fixed energy are superpositions of plane waves, with wavevectors on hyperbolas rather than circles. The local velocity (e.g. in the Madelung representation) is proportional to the kinetic momentum, i.e. local particle velocity, not the more familiar canonical momentum (phase gradient). The patterns of the associated streamlines are different, especially near phase singularities and phase saddles where the kinetic and canonical streamline patterns have opposite indices. Contrasting with the superficially analogous circular smooth solutions of kinetically isotropic Hamiltonians are wave modes that are anisotropic in position and also discontinuous. Pictures illustrating these phenomena are included. The occurrence of familiar concepts in unfamiliar guises could be useful for teaching quantum or wave physics at graduate level.

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“…See the example in section 5.10. The properties of the quantized vortices that form at nodes in the fluid density, and their relation to the Aharonov-Bohm effect, have been the studied somewhat extensively [19,25,[34][35][36][37][38][39][40][41].…”

Section: Superoscillations In Energy Eigenstatesmentioning

confidence: 99%

“…It is reasonable to say that all of the apparently nonclassical behavior of this fluid is due to the role of the quantum potential energy, and if this is interpreted as a new type of classical potential energy, then the behavior of the fluid is entirely classical. There has been some recent renewed interest in understanding the role and applications of the quantum potential [20][21][22][23][24][25][26][27][28]. Importantly, there is also a relatively new local quantum formalism [29] which replaces the configuration space wavefunction with a set of single-particle wavefunctions in spacetime, so even entangled many-particle systems can be understood using this fluid picture.…”

Section: Introductionmentioning

confidence: 99%

Madelung mechanics and superoscillations

Waegell

2024

New J. Phys.

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In single-particle Madelung mechanics, the single-particle quantum state $\Psi(\vec{x},t) = R(\vec{x},t) e^{iS(\vec{x},t)/\hbar}$ is interpreted as comprising an entire conserved fluid of classical point particles, with local density $R(\vec{x},t)^2$ and local momentum $\vec{\nabla}S(\vec{x},t)$ (where $R$ and $S$ are real). The Schr"{o}dinger equation gives rise to the continuity equation for the fluid, and the Hamilton-Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term $Q(\vec{x},t) = -\frac{\hbar^2}{2m}\frac{\vec{\nabla}R(\vec{x},t)}{R(\vec{x},t)}$, which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of $\Psi$ exceeds its global band limit. Berry showed that for states of definite energy $E$, the regions of superoscillation are exactly the regions where $Q(\vec{x},t)<0$. For energy superposition states with band-limit $E_+$, the situation is slightly more complicated, and the bound is no longer $Q(\vec{x},t)<0$. However, the fluid model provides a definite local energy for each fluid particle which allows us to define a local band limit for superoscillation, and with this definition, all regions of superoscillation are again regions where $Q(\vec{x},t)<0$ for general superpositions. An alternative interpretation of these quantities involving a \textit{reduced quantum potential} is reviewed and advanced, and a parallel discussion of superoscillation in this picture is given. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.

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Time-independent, paraxial and time-dependent Madelung trajectories near zeros

Cited by 4 publications

References 38 publications

Classical trajectories from the zeros of the quantum potential: the 2D isotropic harmonic oscillator

Classical trajectories from the zeros of the quantum potential: the 2D isotropic harmonic oscillator

Kinetically anisotropic Hamiltonians: plane waves, Madelung streamlines and superpositions

Madelung mechanics and superoscillations

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