81–87 of 87 posts
Feynman wrote in his autobiography that much of his success came from having different mathematical tricks than most of his peers. So when they were stuck, he could sometimes make progress.
Feynman used to read his own books. When asked he said, "it's all in here". He used to revise and refresh his own understanding.
From a comment: > These methods can be combined. First generalize the problem, making it more complicated. Then simplify along a different axis. – Stig Hemmer Very relevant to software design too.
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>Success is relative. If the goal is to never fail, never try is the best strategy. This is dangerously not true, if you never try, then you are guaranteed to fail to live up to your potential, which is one of the greatest failures of all.
Sorry if it wasn't clear and explicit enough, but this wasn't meant to be an advise to not try anything. Just an extrem example of what it might be logically concluded when starting from unsound grounds. Never fail is not a sound goal. That said living up to our potential is not very much better per se. Or at least it seems far too vague to be aptly considered. What forces are going to define what it is to tap releva…
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If you know some basic composition methods, Euler, RK and all the higher order methods can be easily constructed as you need them. But there are still many applications where you for example want to use symplectic methods instead. If you know about those+composition, you can at least solve more or less all of classical newtonian dynamics. But if you go to quantum mechanics or field theory (or quantum field theory), y…
PDEs are PDEs, regardless of where they come from Newtonian or quantum. Would you care to elaborate why you think quantum requires a new kind of numerical analysis?
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There's a reason why numeric analysis is still actively studied by research mathematicians. If we could just throw something as simple as newton's method at any nonlinear problem, we'd only need people to learn this once in school and everyone could solve everything.
But I'm not sure I'd recommend going into the field. There's something demoralizing about doing research on something which already has dozens of valid and successful methods, of which you are trying to create a slightly more optimized version.
If you ever get a PhD, you'll find that this is pretty much all of academia.
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PDEs are PDEs, regardless of where they come from Newtonian or quantum. Would you care to elaborate why you think quantum requires a new kind of numerical analysis?
For field theory, you can still go some way using normal finite-difference approaches, but you have entered a huge can of worms regarding stability. For quantum physics, the problem starts well before you even get to writing a solver, since (at least for QFT) you are actually dealing with operator-valued distributions rather than normal fields - and that in extremely high (even infinite) dimensional spaces. That mean…
Earlier quoted context omitted.
For field theory, you can still go some way using normal finite-difference approaches, but you have entered a huge can of worms regarding stability. For quantum physics, the problem starts well before you even get to writing a solver, since (at least for QFT) you are actually dealing with operator-valued distributions rather than normal fields - and that in extremely high (even infinite) dimensional spaces. That mean…
The challenges you mentioned, and techniques to address them, are not unique to quantum physics. I am still not understanding how quantum physics require "new" kind of numerical analysis. And what are these new kinds of techniques you hint at? Could you give me some examples of unique techniques that arose from quantum physics and are not used elsewhere?