Probability and complex quantum trajectories

John, Moncy V.

doi:10.1016/j.aop.2008.09.007

Cited by 43 publications

(34 citation statements)

References 30 publications

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“…As in the previous case, the MdBB equations of motion is given by (16), but with (x, y, z) as complex variables. The partial derivatives to be used in these equations are, respectively, given by equations (12), (13) and (14).…”

Section: Interference Pattern In the Mdbb Approach -Stationary Statementioning

confidence: 99%

“…It may be noted that dBB is not the only quantum trajectory formalism available in the literature. The Floyd, Faraggi and Matone (FFM) [10] and the modified de Broglie-Bohn (MdBB) [11,12,13,14,15] trajectory representations have also received wide attention in recent years. The equations of motion used in dBB and MdBB schemes are alike, of the general form mṙ i = ∇ i S, where S represents the Hamilton-Jacobi functions in the respective quantum Hamilton-Jacobi equations in the two schemes.…”

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confidence: 99%

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Interfering Quantum Trajectories Without Which-Way Information

Mathew¹,

John²

2017

Found Phys

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Quantum trajectory-based descriptions of interference between two coherent stationary waves in a double-slit experiment are presented, as given by the de Broglie-Bohm (dBB) and modified de Broglie-Bohm (MdBB) formulations of quantum mechanics. In the dBB trajectory representation, interference between two spreading wave packets can be shown also as resulting from motion of particles. But a trajectory explanation for interference between stationary states is so far not available in this scheme. We show that both the dBB and MdBB trajectories are capable of producing the interference pattern for stationary as well as wave packet states. However, the dBB representation is found to provide the 'which-way' information that helps to identify the hole through which the particle emanates. On the other hand, the MdBB representation does not provide any which-way information while giving a satisfactory explanation of interference phenomenon in tune with the de Broglie's wave particle duality. By counting the trajectories reaching the screen, we have numerically evaluated the intensity distribution of the fringes and found very good agreement with the standard results.

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Section: Interference Pattern In the Mdbb Approach -Stationary Statementioning

confidence: 99%

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confidence: 99%

Interfering Quantum Trajectories Without Which-Way Information

Mathew¹,

John²

2017

Found Phys

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“…A clear advantage of the MdBB complex representation over other trajectory formalisms is that in it the Born probability distribution along the real line can be directly obtained from the velocity field [23]. It is seen that an exponential function involving the integral of the imaginary part of the MdBB velocity field provides this distribution in such a way that on the real line, the more we are inside the closed trajectories, the larger is the probability density.…”

Section: Introductionmentioning

confidence: 99%

Tunneling in energy eigenstates and complex quantum trajectories

Mathew¹,

John²

2015

Quantum Stud.: Math. Found.

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Complex quantum trajectory approach, which arose from a modified de Broglie-Bohm interpretation of quantum mechanics, has attracted much attention in recent years. The exact complex trajectories for the Eckart potential barrier and the soft potential step, plotted in a previous work, show that more trajectories link the left and right regions of the barrier, when the energy is increased. In this paper, we evaluate the reflection probability using a new ansatz based on these observations, as the ratio between the total probabilities of reflected and incident trajectories. While doing this, we also put to test the complex-extended probability density previously postulated for these quantum trajectories. The new ansatz is preferred since the evaluation is solely done with the help of the complex-extended probability density along the imaginary direction and the trajectory pattern itself. The calculations are performed for a rectangular potential barrier, symmetric Eckart and Morse barriers, and a soft potential step. The predictions are in perfect agreement with the standard results for potentials such as the rectangular potential barrier. For the other potentials, there is very good agreement with standard results, but it is exact only for low and high energies. For moderate energies, there are slight deviations. These deviations result from the periodicity of the trajectory pattern along the imaginary axis and have a maximum value only as much as 0.1 % of the standard value. Measurement of such deviation shall provide an opportunity to falsify the ansatz.

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“…This author proposed a timedependent complex quantum trajectory formalism (based on the same connection formula mentioned by Leacock and Padgett) to study the dynamics associated with some simple analytical cases, such as the harmonic oscillator or the step barrier. Later on this modified de BroglieBohm approach, as denoted by John, has also been applied to the analysis of the Born rule and the normalization conditions of the probability density in the complex plane [374,375], or the dynamics of coherent states [376,377]. Analogous studies carried out to determine different dynamical properties with this complex Bohmian representation have been carried out extensively in the literature [349,[378][379][380][381][382][383][384][385][386][387][388][389][390][391][392][393][394][395][396][397].…”

Section: Trajectories From Complex Actionmentioning

confidence: 99%

Applied Bohmian Mechanics

Oriols¹,

Mompart²

2012

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Abstract. Bohmian mechanics provides an explanation of quantum phenomena in terms of point-like particles guided by wave functions. This review focuses on the use of nonrelativistic Bohmian mechanics to address practical problems, rather than on its interpretation. Although the Bohmian and standard quantum theories have different formalisms, both give exactly the same predictions for all phenomena. Fifteen years ago, the quantum chemistry community began to study the practical usefulness of Bohmian mechanics. Since then, the scientific community has mainly applied it to study the (unitary) evolution of single-particle wave functions, either by developing efficient quantum trajectory algorithms or by providing a trajectorybased explanation of complicated quantum phenomena. Here we present a large list of examples showing how the Bohmian formalism provides a useful solution in different forefront research fields for this kind of problems (where the Bohmian and the quantum hydrodynamic formalisms coincide). In addition, this work also emphasizes that the Bohmian formalism can be a useful tool in other types of (nonunitary and nonlinear) quantum problems where the influence of the environment or the nonsimulated degrees of freedom are relevant. This review contains also examples on the use of the Bohmian formalism for the many-body problem, decoherence and measurement processes. The ability of the Bohmian formalism to analyze this last type of problems for (open) quantum systems remains mainly unexplored by the scientific community. The authors of this review are convinced that the final status of the Bohmian theory among the scientific community will be greatly influenced by its potential success in those types of problems that present nonunitary and/or nonlinear quantum evolutions. A brief introduction of the Bohmian formalism and some of its extensions are presented in the last part of this review.

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Probability and complex quantum trajectories

Cited by 43 publications

References 30 publications

Interfering Quantum Trajectories Without Which-Way Information

Interfering Quantum Trajectories Without Which-Way Information

Tunneling in energy eigenstates and complex quantum trajectories

Applied Bohmian Mechanics

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