# Revisiting Quantum Notions of Stress

I plan to submit the attached paper on quantum mechanical definition of stress in the next few weeks. Comments and feedback are welcome. Fair amount of work has been done on stress definition in the context of classical molecular dynamics (also attracting some controversies). In contrast, there appear to be several open issues in the quantum case. Hopefully, the attached paper provides a starting point.

Abstract: An important aspect of multiscale modeling of materials is to link continuum concepts such as fields to the underlying discrete microscopic behavior in a seamless manner. With the growing importance of atomistic calculations to understand material behavior, reconciling continuum and discrete concepts is necessary to interpret molecular and quantum mechanical simulations. In this work, we provide a quantum mechanical framework to a distinctly continuum quantity: mechanical stress. While the concept of the global macroscopic stress tensor in quantum mechanics has been well established, there still exist open issues when it comes to a spatially varying local quantum stress tensor. We attempt to shed some light on this topic by establishing a general quantum mechanical operator based approach to continuity equations and from those, introduce a local quantum mechanical stress tensor. Further, we elucidate the analogies that exist between (classical) molecular dynamics based stress definition and the quantum stress. Our derivations seem to suggest that the local quantum mechanical stress may not be an observable in quantum mechanics and therefore traces the non-uniqueness of the atomistic stress tensor to the gauge arbitrariness of the quantum mechanical state-function. Lastly, the virial stress theorem (of empirical molecular dynamics) is re-derived in a transparent manner that elucidates the analogy between quantum mechanical global stress.

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## Re: Pradeep's paper

Dear Pradeep:

Interesting work and I'm glad it's finally completed. I have some minor comments.

1) In Eq.(12), you don't define T.

2) In Eq.(17), can \epslion_{\alpha\beta} be non-symmetric?

3) In Eq. (34), you motivate/define stress given the force f. This is something that has been considered by many as you mention throughout the paper. Again, as you mention, this is used in electromagnetism to define Maxwell's "stress" tensor. I don't have any objections but is this what we do in continuum mechanics? I think the answer is no. We start with traction, something that represents the mutual mechanical interaction of two pieces of the body in contact. Then as a consequence of balance of linear momentum Cauchy's theorem is proved; a stress tensor exists (traction is linear in unit normal vector and given the unit normal, stress tensor is the operator acting on the unit normal giving the traction). In other words, we don't start with closed surfaces and the total force acting on them; interaction of two bodies in contact on a (small and not closed) surface is considered. In electromagnetism, the total force acting on a subbody is obtained and then one looks for something divergence of which is the force. I'm not sure if Maxwell "stress" is a real stress in the sense of continuum mechanics, though it can be a useful quantity. In the discrete setting, it is true that there is this gauge invariance but I'm not sure if this is the best way of approaching the stress problem, though at this time I don't have a better alternative in mind. Just something to think about/discuss.

4) If you're thinking about a mechanics journal, perhaps a short appendix on basics of quantum mechanics will make the paper self-contained.

5) Some minor typos: i) Page 10, six lines before Eq.(34), "stress .field" should read "stress fiend" I think. ii) Page 11, three lines after Eq. (36), "should be chose" should read "should be chosen". iii) The same page right after Eq. (38) I think "and whenever" should be deleted. iv) Page 18, in the last line of the second paragraph "then stress" should read "than stress".

Regards,

Arash

## Arash, thanks very much for

Arash, thanks very much for looking through the paper and for your comments. We (unfortunately) changed symbols around Equation (12) for the stess tensor. I will correct this and the typos you kindly pointed out.

Regarding, epsilon, it can be unsymmetric but we consider a symmetric tensor to avoid worrying about rotation down the road.

Regarding you point #3; unless I have mis-interpreted your remark, I am not quite sure how to avoid the force-stress identity we used or whether starting by the traction route will get us a different answer. I will be curious to know a bit more about your thoughts on this....

I am still unsure of which journal to send this to. Once I have refined a bit further, if a mechanics editor is willing to consider the paper, I will explore with the editor if a Appendix of the type you suggest would be appropriate. I personally think it woudl be good but lately I have had a few such removed by referees as being superfluous.