Bohmian dwell times without trajectories

Within the framework of Bohmian mechanics dwell times find a straightforward formulation. The computation of associated probabilities and distributions however needs the explicit knowledge of a relevant sample of trajectories and therefore implies formidable numerical effort. Here, a trajectory free formulation for the average transmission and reflection dwell times within static spatial intervals [ a , b ] is given for one-dimensional scattering problems. This formulation reduces the computation time to less than 5% of the computation time by means of trajectory sampling.Within the framework of Bohmian mechanics dwell times find a straightforward formulation. The computation of associated probabilities and distributions however needs the explicit knowledge of a relevant sample of trajectories and therefore implies formidable numerical effort. Here, a trajectory free formulation for the average transmission and reflection dwell times within static spatial intervals [ a , b ] is given for one-dimensional scattering problems. This formulation reduces the computation time to less than 5% of the computation time by means of trajectory sampling.

arXiv:quant-ph/0406045Journal of Physics A Mathematical and GeneralMay 25, 20052 citations

Bohmian arrival time without trajectories

Sabine Kreidl, Gebhard Grübl, Hans Georg Embacher

The computation of detection probabilities and arrival time distributions within Bohmian mechanics in general needs the explicit knowledge of a relevant sample of trajectories. Here it is shown how for one-dimensional systems and rigid inertial detectors these quantities can be computed without calculating any trajectories. An expression in terms of the wavefunction Ψ and its spatial derivative ∂ x Ψ, both restricted to the boundary of the detector's spacetime volume, is derived for the general case, where the probability current at the detector's boundary may vary its sign.The computation of detection probabilities and arrival time distributions within Bohmian mechanics in general needs the explicit knowledge of a relevant sample of trajectories. Here it is shown how for one-dimensional systems and rigid inertial detectors these quantities can be computed without calculating any trajectories. An expression in terms of the wavefunction Ψ and its spatial derivative ∂ x Ψ, both restricted to the boundary of the detector's spacetime volume, is derived for the general case, where the probability current at the detector's boundary may vary its sign.

arXiv:quant-ph/0305163Journal of Physics A Mathematical and GeneralAug 5, 200319 citations

Implications of the noncrossing property of Bohm trajectories in one-dimensional tunneling configurations

Xavier Oriols, Ferran Martı́n, Jordi Suñé

Several practical implications of the noncrossing property of one-dimensional Bohm trajectories are examined. It is shown that the position of a Bohm particle, the average transmission, reflection and dwell times, and the probability distribution of these tunneling times, can all be obtained without actually calculating trajectories. On the other hand, the intuitive interpretation of the scattering of wave packets by potential barriers is discussed within the framework of Bohm's interpretation of quantum mechanics. In this regard, claims that Bohm's approach leads to counterintuitive results are shown to be subjective. \textcopyright{} 1996 The American Physical Society.Several practical implications of the noncrossing property of one-dimensional Bohm trajectories are examined. It is shown that the position of a Bohm particle, the average transmission, reflection and dwell times, and the probability distribution of these tunneling times, can all be obtained without actually calculating trajectories. On the other hand, the intuitive interpretation of the scattering of wave packets by potential barriers is discussed within the framework of Bohm's interpretation of quantum mechanics. In this regard, claims that Bohm's approach leads to counterintuitive results are shown to be subjective. \textcopyright{} 1996 The American Physical Society.

Physical Review AOct 1, 199660 citations

Dwell Times, Wavepacket Dynamics, and Quantum Trajectories for Particles with Spin 1/2

Bill Poirier, Richard Lombardini

The theoretical connections between quantum trajectories and quantum dwell times, previously explored in the context of 1D time-independent stationary scattering applications, are here generalized for multidimensional time-dependent wavepacket applications for particles with spin 1/2. In addition to dwell times, trajectory-based dwell time distributions are also developed, and compared with previous distributions based on the dwell time operator and the flux–flux correlation function. Dwell time distributions are of interest, in part because they may be of experimental relevance. In addition to standard unipolar quantum trajectories, bipolar quantum trajectories are also considered, and found to relate more directly to the dwell time (and other quantum time) quantities of greatest relevance for scattering applications. Detailed calculations are performed for a benchmark 3D spin-1/2 particle application, considered previously in the context of computing quantum arrival times.The theoretical connections between quantum trajectories and quantum dwell times, previously explored in the context of 1D time-independent stationary scattering applications, are here generalized for multidimensional time-dependent wavepacket applications for particles with spin 1/2. In addition to dwell times, trajectory-based dwell time distributions are also developed, and compared with previous distributions based on the dwell time operator and the flux–flux correlation function. Dwell time distributions are of interest, in part because they may be of experimental relevance. In addition to standard unipolar quantum trajectories, bipolar quantum trajectories are also considered, and found to relate more directly to the dwell time (and other quantum time) quantities of greatest relevance for scattering applications. Detailed calculations are performed for a benchmark 3D spin-1/2 particle application, considered previously in the context of computing quantum arrival times.

EntropyApr 14, 20241 citation

Reformulation of Bohmian mechanics

Moise Borilla Licea

Bohmian mechanics as formulated originally in 1952, has been useful in the implementation of numerical methods applied to quantum mechanics. The scientific community though has had ever since a critical thought about it. Therefore, there are still points to be clarified and rectified. The two main problems are basically: Bohmian mechanics gives a privilege role to the position representation. Secondly, the current interpretation of Bohmian trajectories has been recently proven wrong. In this context, in Chapter 2, new complex Bohmian quantities are defined; so that they allow the capacity to formulate Bohmian mechanics in any arbitrary continuous representation, for instance, the momentum representation. This Chapter is fully based on two articles, regarding the proposed complex Bohmian formulation and its extension into momentum space. Chapter 3 deals with a redefinition and reinterpretation of the Bohmian trajectories from the handling of the continuity equation, this is done without any need of additional postulates or interpretations. Also, it is proved that Bohmian mechanics is actually more than a projective aspect of the Wigner function. As a third point, Chapter 4 presents a sytematic treatment of the hydrodynamic scheme of Bohmian mechanics. Then, a brief summary of the transport equations in Bohmian mechanics is done. Next, a unified hydrodynamic treatment is found for the Bohmian mechanics. This treatment is useful to sketch, a Bohmian treatment to efficiently find the steady value of the transmission integral. In Chapter 5 conclusions of this thesis are drawn.Bohmian mechanics as formulated originally in 1952, has been useful in the implementation of numerical methods applied to quantum mechanics. The scientific community though has had ever since a critical thought about it. Therefore, there are still points to be clarified and rectified. The two main problems are basically: Bohmian mechanics gives a privilege role to the position representation. Secondly, the current interpretation of Bohmian trajectories has been recently proven wrong. In this context, in Chapter 2, new complex Bohmian quantities are defined; so that they allow the capacity to formulate Bohmian mechanics in any arbitrary continuous representation, for instance, the momentum representation. This Chapter is fully based on two articles, regarding the proposed complex Bohmian formulation and its extension into momentum space. Chapter 3 deals with a redefinition and reinterpretation of the Bohmian trajectories from the handling of the continuity equation, this is done without any need of additional postulates or interpretations. Also, it is proved that Bohmian mechanics is actually more than a projective aspect of the Wigner function. As a third point, Chapter 4 presents a sytematic treatment of the hydrodynamic scheme of Bohmian mechanics. Then, a brief summary of the transport equations in Bohmian mechanics is done. Next, a unified hydrodynamic treatment is found for the Bohmian mechanics. This treatment is useful to sketch, a Bohmian treatment to efficiently find the steady value of the transmission integral. In Chapter 5 conclusions of this thesis are drawn.

Jan 1, 20210 citations

Negative probability and the distributions of dwell, transmission, and reflection times for quantum tunneling

D. Sokolovski, J. N. L. Connor

We have solved the problem of determining the traversal time distribution for a quantum transition between arbitrary initial and final states. This distribution is given by a Fourier transform of the Feynman averaged characteristic function for the classical traversal time functional ${\mathit{t}}_{\mathrm{ab}}^{\mathrm{cl}}$ [x(t)]. For one-dimensional tunneling, the distributions for the dwell, reflection, and transmission times are determined for an arbitrary potential V(x). All three distributions are complex valued and require the concept of negative probability.We have solved the problem of determining the traversal time distribution for a quantum transition between arbitrary initial and final states. This distribution is given by a Fourier transform of the Feynman averaged characteristic function for the classical traversal time functional ${\mathit{t}}_{\mathrm{ab}}^{\mathrm{cl}}$[x(t)]. For one-dimensional tunneling, the distributions for the dwell, reflection, and transmission times are determined for an arbitrary potential V(x). All three distributions are complex valued and require the concept of negative probability.

Physical Review AAug 1, 199162 citations

{\mathit{t}}_{\mathrm{ab}}^{\mathrm{cl}}

[x(t)]. For one-dimensional tunneling, the distributions for the dwell, reflection, and transmission times are determined for an arbitrary potential V(x). All three distributions are complex valued and require the concept of negative probability.