Equal Radiation Frequencies from Different Transitions in the Non-Relativistic Quantum Mechanical Hydrogen Atom

Round 1

Reviewer 1 Report

This is a fun paper and it should be published subject to some light corrections.

The most urgent is to update/amend reference 2: in its current form it is uninformative and not enough details are provided to access or recover the material.  In addition, the text above Eq.(15) refers to author Perondi but [2] is a citation to author Kantor.  A similar general comment as to accessibility applies to reference 1.   It is not clear why a reference to the Pell equation is needed (reference [6]) so maybe the authors could expand (if warranted) on the relevance of this equation in physics.

As I suspect the topic of number theory will not be that familiar to most users,  some additional references would be welcome and details could be provided in the derivations.  

The syntax can also be improved (v.g. emergent-> emergence) but this is minor and the current text does not require major English revisions. 

Author Response

\textbf{This is a fun paper and it should be published subject to some light corrections.}

We thank the reviewer for the kind comment.

\ \ 

\textbf{The most urgent is to update/amend reference 2: in its current form it is uninformative and not enough details are provided to access or recover the material. In addition, the text above Eq.(15) refers to author Perondi but [2] is a citation to author Kantor. A similar general comment as to accessibility applies to reference 1.} 

The problem was discussed in these references, with many possible directions to explored (and credited to people who give the suggestions). We have fixed and added the Links to these references, making it more accessible to readers. 

\ \ 

\textbf{It is not clear why a reference to the Pell equation is needed (reference [6]) so maybe the authors could expand (if warranted) on the relevance of this equation in physics.}

We agree with the reviewer that, in a sense, the reference might be redundant. However, we find it important to point out similarity for a more math-oriented readers (Pell equation is indeed very well-known in number theory, has been mentioned in all textbooks of the subject), since provoking analogy does wonders to physicists. We decide to keep the reference. We hope for your understanding.

\ \ 

\textbf{As I suspect the topic of number theory will not be that familiar to most users,  some additional references would be welcome and details could be provided in the derivations.}

We agree with this suggestion. We have added a few more references to the manuscript, especially on the concepts that (we feel) are familiar to number theorists but not physicists. Also a lot more references, in general.

\ \ 

\textbf{The syntax can also be improved (v.g. emergent-> emergence) but this is minor and the current text does not require major English revisions.}

We have corrected the mistakes, and have gone through the manuscript more carefully.

Author Response File: Author Response.pdf

Reviewer 2 Report

I recommend publication of this manuscript as a curious, but unimportant puzzle of the nonrelativistic hydrogen spectrum.  The authors should mention that they are dealing with the NONRELATIVISTIC hydrogen atom. and should ask for proofreading by a native English speaker.  The last line of the abstract requires an "is."  The line above Eq. (3) requires a "which," altering the meaning of the sentence.  The top line above Eq. (18) has "the" repeated. etc. 

Author Response

\textbf{I recommend publication of this manuscript as a curious, but unimportant puzzle of the nonrelativistic hydrogen spectrum.}

In a sense, we agree with the reviewer. While the puzzle might not be important to fundamental studies of hydrogen spectral, we find it interesting to realize that knowing the radiation frequency is not enough to determine the transition.

\ \ 

\textbf{The authors should mention that they are dealing with the NONRELATIVISTIC hydrogen atom.} 

We did mention that in the Introduction, but we now realize that it was not very clear. We have change the title of the paper (and also add the non-relativistic description in the Abstract and Figures).

\ \ 

\textbf{and should ask for proofreading by a native English speaker.}

We have checked the English-issue very carefully this time.

\ \ 

\textbf{The last line of the abstract requires an "is."  The line above Eq. (3) requires a "which," altering the meaning of the sentence.  The top line above Eq. (18) has "the" repeated. etc.}

We also fix those errors, either by direct substitution (as suggested by the reviewer) or by rephrasing the sentence to clarify what we mean.

Author Response File: Author Response.pdf