[Paper Review] Reply to `No Contradictions between Bohmian and quantum mechanics'

Partha Ghose|ArXiv.org|Aug 1, 2000

Quantum Mechanics and Applications1 references3 citations

TL;DR

This paper refutes Marchildon's claim that Bohmian mechanics and standard quantum mechanics are compatible by demonstrating that Marchildon's derivation contains a critical error: he neglects essential approximations required for Fraunhofer diffraction. When these approximations—specifically, the omission of quadratic terms in $ a $ and $ x $—are properly applied, the trajectories remain symmetric and non-crossing, confirming the incompatibility between Bohmian mechanics and standard quantum mechanics as originally claimed.

ABSTRACT

Marchildon's claim (quant-ph/0007068) regarding Ghose's papers (quant-ph/0001024 and 0003037) is shown to be erroneous.

Motivation & Objective

• To correct Marchildon's misinterpretation of the conditions under which Bohmian trajectories are symmetric and non-crossing.
• To reaffirm the incompatibility between Bohmian mechanics and standard quantum mechanics under physically valid approximations.
• To demonstrate that the symmetry of trajectories arises from the Fraunhofer diffraction approximation, not from translation invariance alone.
• To clarify that Marchildon's equations reduce to zero under proper physical constraints, invalidating his conclusion of compatibility.

Proposed method

• Re-expresses Marchildon's equation (18) in terms of $ y = x_1 + x_2 $, showing $ y(t) = y(0)e^{ct} $, and since $ y(0) = 0 $, $ y(t) = 0 $ for all $ t $, implying symmetry.
• Applies the Fraunhofer diffraction approximation by dropping quadratic terms in $ a $ and $ x $ from Marchildon's equations (13) and (14).
• Derives corrected expressions for $ r_A \approx y - \frac{2ax}{2L} $ and $ r_B \approx y + \frac{2ax}{2L} $, which are substituted into equation (17).
• Shows that under these approximations, the right-hand side of Marchildon's equation (18) vanishes, implying no net force or asymmetry.
• Argues that the symmetry of trajectories is a consequence of the diffraction approximation, not translation invariance, and thus the incompatibility remains valid.
• Asserts that the conditions used in the author's original papers are sufficient but not necessary for the result, reinforcing the core conclusion.

Experimental results

Research questions

• RQ1 Does the assumption of translation invariance in Marchildon's model correctly establish compatibility between Bohmian and standard quantum mechanics?
• RQ2 What happens to the trajectory symmetry when the Fraunhofer diffraction approximation is properly applied?
• RQ3 Are the quadratic terms in $ a $ and $ x $ in Marchildon's equations physically negligible under the intended physical conditions?
• RQ4 Can the symmetry and non-crossing of Bohmian trajectories be derived without assuming translation invariance?
• RQ5 Is the incompatibility between Bohmian mechanics and standard quantum mechanics still valid when the correct approximations are applied?

Key findings

• Marchildon's equation (18) leads to $ y(t) = 0 $ for all $ t $, confirming trajectory symmetry, but this result is contingent on the initial condition $ y(0) = 0 $, not on translation invariance.
• When the Fraunhofer diffraction approximation is applied, the quadratic terms in $ a $ and $ x $ must be neglected, reducing Marchildon's equations (13) and (14) to linear forms.
• The corrected expressions for $ r_A $ and $ r_B $ lead to a vanishing right-hand side in equation (18), implying no asymmetry and confirming trajectory symmetry.
• The symmetry of trajectories is thus a consequence of the diffraction approximation, not translation invariance, validating the author's original conclusion.
• The conditions used in the author's original work are sufficient but not necessary for the incompatibility result, reinforcing the robustness of the conclusion.
• Marchildon's claim of compatibility is invalidated because it fails to account for the physical constraints required for the diffraction regime.