Discovering Bohr’s Yin-Yang Diagram in Quantum Tunneling Dynamics

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In this paper the author shows that pairs of quantum variables can be represented as diagrams resembling the Ying Yang symbol.  As pairs of quantum variables,  the author considers: (1) The visibility of photon wave fringes and the which-way distinguishability  in a Mach-Zenhder interferometer. (2) The real and imaginary parts of the complex velocity of a particle  that tunnels into a potential step. (3) The real and imaginary parts of the particle's position in the potential step. The latter two pairs appear in a complex-trajectory formulation of quantum mechanics described by the author.

The author connects the Ying Yang-like diagrams to Bohr's principle of complementarity, and points out that, interestingly,  Bohr had used a Ying Yang-like symbol in his coat of arms to represent complementarity. 

It's not clear to me if the results of the paper are new, because I found out that the author has considered similar results in previous publications. However, it is an interesting, memorable presentation for people like me, who were not familiar with the author's formulation.  The paper can spark the interest of other people in the complex trajectory formulation of quantum theory and its connection to complementarity.

The author should address the following points before I  recommend the publication of the paper:

1) In line 273, the author says that the quantum YYD in the Mach-Zehnder interferometer has a "slight difference" from the Taoist YYD. However, I think the difference is significant (quadratic vs. linear). As far as I understand, the quadratic form in the quantum YYD comes from the quantum principle that probabilities are squares of wave functions, whereas in classical probabilities this concept does not exist.  Please clarify this point. 

2) The author claims that, in the tunneling process, the imaginary part of the particle's velocity quantifies the degree of the particle's wave nature and the "uncertainty in its motion". The paper has a reference to one the author's previous publications, but I could not have free access to this publication. Can the author please add a short explanation of why the imaginary part of the particle's velocity quantifies the degree of the particle's wave nature, after line 524?

3) The following question is probably related to my previous question: In line 534, the author says that when the particle exits the tunneling region, its wave nature  disappears completely and in line 558 the author says that the particle exhibits pure particle nature then. What does this mean? In the standard quantum mechanical treatment of the potential step, the particle always has an uncertainty in position and it always has a wave  function (as long as the wave function has not collapsed). Am I misunderstanding? Please explain.

4) The author has shown that the real and imaginary velocities do not form ideal YYDs for small "n". Can the author explain why is this so? If these variables are  not complementary pairs for small n, why do they become complementary when n is large?

6) Why are the complex-position pA and pB given by Eq. (49)?

Comments on the Quality of English Language

The paper needs editing to improve the English grammar.

In lines 41 and 72, there should be no "the" before "Bohr's".

In line 83 it should say "freedom", not "freedoms".

In line 85 it shouls say "splitters", not "splitter".

In lines 85-94 sometimes it says MZI and other times it says ZMI. Is this correct?

In line 370 I believe the "x" are missing a dot above them.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The aim of the paper is to clarify why Bohr adopted a Ying-Yang diagram (YYD) in his coat of arms.  This paper proposes a definition of a YYD that can be calculated by using complementary variables such as e.g. the real and imaginary part of the momentum of a tunnelling particle. By considering such a particle in a step potential, the authors carry out a tour de force of Bohmian mechanics using complex trajectories. A YYD that looks qualitatively similar to Bohr's is obtained. Also another computation based on an interference experiment is done with resulting YYD further apart from that of Bohr. While the paper is well-written, it has some weak sides. First, the construction of a YYD seems far from unique(?): It depends on the potential, the choice of complementary variables, tunnelling intensity etc. Moreover, Bohr's YYD is, as the author also notes, based on a simple mathematical construction that has nothing to do with quantum mechanics(?). Thus these two diagrams can only be expected to agree in a very qualitative way. Is that satisfactory? Maybe?

Probably only few people would be interested in a way of mapping the quantum mechanics of a system into a YY diagram showing the degree of complementarity of two of its variables? To be critical: what can the YYD of a system be used for - besides being a piece of (simple) art? It does not even seem to illuminate the physics of the system in any intuitive way. However, I think anyway that a demonstration that the YYD of Bohr can - at least qualitatively, be reconstructed through the dynamics of complementary variables is worth publishing. Entropy is a journal that publish papers which are not always main-stream and since this manuscript is well written with detailed and illuminating calculations, I recommend its publication.  

The author should however provide some discussion and more examples of complementary variables in quantum mechanical systems. The complementary principle of Bohr is not very well explained; this should be fixed. Also, the manuscript would be more readable for the non-specialists if qualitative arguments are given why imaginary momentum makes a particle more wave-like.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The author has answered all my questions and I am satisfied with the answers. I thank the author for explaining his previous work, which helped  me understand better the current paper. I recommend the paper for publication.

Comments on the Quality of English Language

The author made the corrections I suggested. Overall, I find the quality of English Language is acceptable.

Reviewer 2 Report

Comments and Suggestions for Authors

publish as is