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I think a lot about the implication of QM. I don't need to understand the mathematical depth, just like I don't need to understand how a computer processes at t

I think a lot about the implication of QM. I don't need to understand the mathematical depth, just like I don't need to understand how a computer processes at the lowest level to do computer science. The point the author makes is VERY interesting in that many, many people really only see the concepts at face value. And that extends into many domains. But I get that not everyone is a dreamer ;)

defterGoose 6y ago
What's most interesting to me is that I never grok a piece of physics without understanding the math (maybe Newtonian mechanics is a gimme because of the way our brains are built), but the math is also rarely part of the "intuition" of things, which is what i strive for in learning. I think this is where people really get into trouble with QM. The physical reality of it is so strange and counterintuitive, that maybe the average brain intuits that the math can't possibly make sense either. When i took quantum in college, i knew i wasnt really "getting it" like others in my class. Yes, i could describe the results of a slit experiment, or in what ways wave-particle duality is a thing, but my brain never opened that "Aha!" gate that it had so many times before. In retrospect, I can say that my unfamiliarity with more advanced algebra and a lack of understanding of math in a more set-theoretical way held me back. As i would move into calculation, i understood in theory how to do the calculus and such, but my brain kept nagging at me with "yeah, but you dont understand why the integral is this way", which would stymie further progress. I found that a big "aha" moment came when i finally understood the relationship between matrices, vector spaces, operators, and functions in the infinite-dimensional case (or even the finite one really). I think i understand now that my difficulty in learning it before was based on my misunderstanding of the difference between representing functions as a formula vs their representation as a vector. I was confused about why you'd need infinite information in a vector to describe something like y=cos(x) which is clearly a finite representation. The answer is that the 'named' functions and simple polynomials you're used to dealing with are represented by infinite vectors...with almost all coefficients as zero. I think was only able to get to this place of (still limited) understanding by putting the QM book down and immersing myself in set theory. In my opinion, we do kids a disservice by not introducing abstract algebra earlier. It would be helpful i think to maybe throw some of that into Precalc, even if its only some basics about the concepts of sets and maps between them.

defterGoose 6y ago

What's most interesting to me is that I never grok a piece of physics without understanding the math (maybe Newtonian mechanics is a gimme because of the way our brains are built), but the math is also rarely part of the "intuition" of things, which is what i strive for in learning. I think this is where people really get into trouble with QM. The physical reality of it is so strange and counterintuitive, that maybe the average brain intuits that the math can't possibly make sense either. When i took quantum in college, i knew i wasnt really "getting it" like others in my class. Yes, i could describe the results of a slit experiment, or in what ways wave-particle duality is a thing, but my brain never opened that "Aha!" gate that it had so many times before. In retrospect, I can say that my unfamiliarity with more advanced algebra and a lack of understanding of math in a more set-theoretical way held me back. As i would move into calculation, i understood in theory how to do the calculus and such, but my brain kept nagging at me with "yeah, but you dont understand why the integral is this way", which would stymie further progress. I found that a big "aha" moment came when i finally understood the relationship between matrices, vector spaces, operators, and functions in the infinite-dimensional case (or even the finite one really). I think i understand now that my difficulty in learning it before was based on my misunderstanding of the difference between representing functions as a formula vs their representation as a vector. I was confused about why you'd need infinite information in a vector to describe something like y=cos(x) which is clearly a finite representation. The answer is that the 'named' functions and simple polynomials you're used to dealing with are represented by infinite vectors...with almost all coefficients as zero. I think was only able to get to this place of (still limited) understanding by putting the QM book down and immersing myself in set theory. In my opinion, we do kids a disservice by not introducing abstract algebra earlier. It would be helpful i think to maybe throw some of that into Precalc, even if its only some basics about the concepts of sets and maps between them.