What has been the most difficult concept for you to grasp?

There are those who think that given the proper surroundings you can grasp any concept. Salmon Khan, someone whose Khan Academy I greatly admire, says that the only thing you have to know is “that you can learn anything.”

He qualifies that somewhat when he adds:
Most people are held back not by their innate ability, but by their mindset. They think intelligence is fixed, but it isn’t. Your brain is like a muscle. The more you use it and struggle, the more it grows. New research shows we can take control of our ability to learn. We can all become better learners. We just need to build our brains in the right way.

Now, “that you can learn anything” has not been my own personal experience. There have been an endless number of things that I have not been able to learn (I’m still struggling with elements of pre-calculus, not to mention power series and intervals of convergence from my Cal 2 text). But let me be quick to say that that’s not the same thing as saying you can’t try, and profitably, to learn anything. And I think that’s probably what Sal means.

Now even those who seem to be the brightest of the brightest, such as, for example, the STEM graduates of Cal Tech, MIT, Stanford, Harvard and Princeton probably find some concepts too difficult for them. And if they don’t while in school they keep at it until they do (although that’s probably not the reason they keep at it).
All this is to say that intelligence and other gifts are not equally shared or distributed. From experience we learn early on that they are not. But this is something that we talk about very little, because it’s politically incorrect to say so in a country such as the United States where we are not all, somehow, “equal.” Nor is it to say that everyone’s intelligence doesn’t have limits, even that of Richard Feynman. It does.
Anyway I stumbled on these ideas while thinking about a Quora question that came in my email this morning, What is the most difficult concept to grasp in physics? and then about Alejandro Jenkins’ fascinating answer that I repeat here:

A famous Harvard physics professor (Ed Purcell maybe?), said that undergraduate physics students come in expecting that the hardest thing they’ll have to learn will be either relativity or quantum mechanics.  Actually, those are the most novel topics (i.e., the ones involving notions that are the most surprising from our ordinary, common-sense perspective).  But the hardest thing that an undergraduate physics students must learn is the classical dynamics of spinning tops (also called, in this context “rigid bodies”).Having taught classical mechanics to advanced undergraduates in physics, I find this to be true.  The following figure, which I’ve taken from chapter VI, sec. 37 of the Mechanics by Landau andLifshitz, shows possible values of the angular momentum vector, in the non-inertial body frame, for a free, asymmetric top.  The ellipsoid is a surface of constant energy, and the closed curves are given by the intersection of that ellipsoid with spheres of various radii, corresponding to different values of the total magnitude of the angular momentum:

This leads to an interesting result about the free asymmetric top, which some people call the “tennis racket theorem“: the top can spin stably about the principal axes with the least (x_1) or the greatest moments of inertia (x_3), but not around the intermediate axis (x_2).  You can demonstrate this by spinning a tennis racket or a ping-pong paddle in the air, as shown here:

If you still don’t believe me that tops can really be such a headache, I suggest looking up the “Poinsot construction”, which even inspired a humorous poemby Prof. David N. Williams, of the U. of Michigan.  Or check out the explicit solutions to the motion of the free asymmetric top in terms of Jacobi elliptic functions (and this is for a free top, mind you, with no net torque acting on it).

The great theoretical physicist James Clerk Maxwell (1831-1879), discoverer of the laws of electrodynamics, wrote that

“To those who study the progress of exact science, the common spinning-top is a symbol of the labours and the perplexities of men.”

Actually, tops can be such a tricky subject to teach that many lecturers tend to gloss over them, especially now that we’re in a rush to get to quantum physics.
Still, even though the classical mechanics of spinning tops can be hard to grasp, it’s perfectly well defined.  The mathematics of (non-relativistic) quantum mechanics is fairly straightforward by comparison, but the interpretation of what the rules of quantum mechanics mean, especially insofar as they concerns the process of measurement, remains quite obscure.  Most physicists are content to compute observable quantities, leaving the interpretation to the philosophers, an attitude captured in a famous dictum, often wrongly attributed to Richard Feynman, to “shut up and calculate”; see N. D. Mermin, “Could Feynman have said this?“, Physics Today 57, 10 (2004).

Things do get pretty hairy when you need a description that’s both quantum and relativistic, which is the regime of high-energy physics.  This requires what’s known as quantum field theory, which is a subject that still presents many conceptual difficulties, despite its great predictive successes.  But quantum field theory is not usually studied at the undergraduate level.

Alejandro Jenkins  Caltech PhD ’06 (physics), Harvard ’01 (Physics and Math)

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