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I went to grad school for chemical engineering a decade ago and part of my frustration with the courses is that the math was never rigorous enough; when I asked

I went to grad school for chemical engineering a decade ago and part of my frustration with the courses is that the math was never rigorous enough; when I asked for clarification, the inconsistencies were glossed over and hand-waved away. The linked article mentions differential forms, which is a perfect example. In engineering courses, these are often introduced without any rigor or formality. They just “appear” as a way of rewriting the equation. What is a “differential”? Who knows. Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way? Who knows. But here are the steps to solve the equation; good luck on the test. Another example is the quantum chemistry class I took (my first introduction to quantum mechanics). The professor mentioned something about how you can take a measurement and collapse the wave function of an electron; the probability of measuring its position at any location in space is non-zero. I followed up with some question about an experiment to confine the wave function collapse to certain regions of space which could be used to transfer information faster than the speed of light, which I thought wasn’t possible. My question was mostly met with a blank stare and something along the lines of “this course doesn’t cover that” before the professor moved on. The most egregious example was a grad-level course on statistical mechanics. The professor mentioned something about the overall wave function of the system being a Slater determinant of individual wave functions. A student (not me) asked why we have wave functions for individual particles and a separate one for the overall system, and which was more “correct” fundamentally? The professor replied that the wave functions for the individual particles were more fundamental, and the overall wave function for the system was just an approximation. I replied, sure, the Slater determinant is a way of enforcing certain constraints that must be met for the whole system, but one of the defining features of quantum mechanics is that the function describing the state of the entire system is generally inseparable; without this we couldn’t have entanglement. The professor dismissed my response as incorrect and said students shouldn’t challenge professors on topics they don’t know anything about. This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results).

sesm 3y ago
How big a system can be while maintaining a quantum entanglement (also called 'quantum coherence') - is still an open question. A very important one, because it affects quantum computers. Experimentally researchers see, that a big enough system can't maintain quantum coherence for long. If you want to know about how this process is described mathematically, search for 'quantum decoherence'. If you want to know about possible physical interpretations, search for 'objective collapse theory'.
BeetleB 3y ago
For those not in the know: DFT = Density Functional Theory
anon291 3y ago
My experience as a math major is that most non-mathematicians, including engineering and chem/phys professors, have a working understand of math and don't think too deep about it. > Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way? Yes. > This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results). Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.
selimthegrim 3y ago
DFT uses the Kohn-Sham approximation which is single particle and is a ground state theory. There’s definitely been research along the lines you bring up in quantum chemistry but not in DFT. You want to look at the Levy-Lieb density and maybe some of Mazziotti’s papers

sesm 3y ago

How big a system can be while maintaining a quantum entanglement (also called 'quantum coherence') - is still an open question. A very important one, because it affects quantum computers. Experimentally researchers see, that a big enough system can't maintain quantum coherence for long. If you want to know about how this process is described mathematically, search for 'quantum decoherence'. If you want to know about possible physical interpretations, search for 'objective collapse theory'.

BeetleB 3y ago

For those not in the know: DFT = Density Functional Theory

anon291 3y ago

My experience as a math major is that most non-mathematicians, including engineering and chem/phys professors, have a working understand of math and don't think too deep about it. > Is there some kind of axiomatic basis for how these symbols can be manipulated in a consistent way? Yes. > This was a guy whose research career was built on dozens of computational chemistry papers that heavily utilized DFT by the way (and by “research”, I mean plug a file with atom coordinates and atom types into the DFT software, run the software, and publish the results). Again, large portions of academia simply stick to their small subfield, publish there, and don't think much about the broader implications, if they think about them at all. This is why so many discoveries are made by new entrants to a field with a slightly different background.

selimthegrim 3y ago

DFT uses the Kohn-Sham approximation which is single particle and is a ground state theory. There’s definitely been research along the lines you bring up in quantum chemistry but not in DFT. You want to look at the Levy-Lieb density and maybe some of Mazziotti’s papers