Interpreting Bohm quantum potentials in Computing quantum waves exactly from classical action

TL;DR

This paper clarifies the role of the Bohm quantum potential in wave construction, showing it can be assumed zero without loss of generality, and compares different initializations affecting the potential's presence.

Contribution

It extends the proof of a key lemma to explicitly include the Bohm quantum potential and explains conditions under which it can be considered zero.

Findings

The Bohm quantum potential can be assumed zero in wave construction.

Different initializations lead to different presence of the Bohm potential.

The overall wave remains independent of the initialization method.

Abstract

The recent arXiv posting [11], commenting on the paper [7], argues that the proof of Lemma 3.1 in [7] is missing the Bohm quantum potential [1, 2] of the Madelung p.d.e. [9]. This short technical note extends the proof of Lemma 3.1 to introduce a Bohm quantum potential explicitly, and then shows why this term can be assumed to be zero in the wave construction, without loss of generality. The continuity p.d.e. and the Hamilton-Jacobi p.d.e., extended by the Bohm potential, are undisputed. However, the actual action and density solutions depend on their initialization at t = 0. In [7], this initialization is motivated by the Feynman kernel [4], which is fundamentally different from the standard initialization of the Madelung solution [9]. This in turn leads to different action and density solutions, and explains why in one case the Bohm quantum potential disappears and in the other does not. The resulting overall wave, however, is independent of this computational initialization.