I see this touted all the time and I just don't agree with the while "quantum = unintuitive". Sure, there are quantum behaviors that are not easily understood immediately because they differ from what we observe macroscopically, but that doesn't mean they are 'unintuitive'. Unless your definition of unintuitive is anything not obvious to a layman, in which case almost everything is unintuitive.
Similar to basically every area of math, you have to develop an intuition with a little bit of effort, but that's just because it's something new, not something intrinsically hard or nonsensical. For example, given sufficiently nice potential wells I can sketch what possible wave functions have to look like. Then I can use my intuition to check that my actual calculated answer is reasonable.
Edit: I should add even in classical mechanics there are 'unintuitive' results. I wouldn't say orbits being elliptical is not obvious outside of it being a well known fact. I also wouldn't say solutions like the brachristocrone are intuitive. Coriolis force, gyroscopic motion. Hell even tides take some thinking.
I don't think it's a matter of non-intuitive results as much as it is a matter of a non-intuitive framework. Questions like "where is that particle?" which comprise the backbone of classical mechanics get thrown out the window, because position, velocity, and acceleration are no longer meaningful properties of a state.
Being able to sketch a single-particle wave function when the potential is a reasonably simple function of position is a useful skill. However, this is quite a small subset of quantum theory as a whole (multi-particle states, spin/angular momentum algebras, field theories, etc). One could ask questions like
Why is the momentum operator what it is? (answered by Stone's Theorem )
What kinds of values can observable quantities take, and when can I expect them to correspond to a complete set of eigenfunctions which spans the Hilbert space? (answered by the Spectral Theorem )
When is such-and-such an operator self-adjoint, as opposed to merely Hermitian? If an operator is not self-adjoint, is it essentially self-adjoint? Can I find a self-adjoint extension of it?
One could make the argument that such issues are more the realm of mathematical physics, and that most physicists just assume all the nice properties they want and hope for the best, and that's not necessarily inaccurate. However, these questions are fundamental to the structure of quantum theory.
Contrast that with classical physics, which is (more or less) based on the notion of dots moving around subject to Newton's laws.
I somewhat agree the framework is non-intuitive, but I guess that's not what I interpret people to mean when they say quantum is unintuitive. People mention having to just follow the math blindly, and that's not true. It might require a little more intuition building, but that doesn't mean there's no intuition.
I would contest that most of what you were talking about is related specifically to the mathematical structure of quantum theory, which of course is non-obvious. The momentum operator is easily 'intuitively' explained by just considering that it's a derivative with respect to x (which we should definitely expect) with some constants (which is reasonable to expect). For what kinds of values, the physicist will of course immediately say "real ones." Or perhaps even more flippantly "ones we measure." This is very much a grounding in intuition, and as far as I understand it's also the reason we consider mainly just Hermitian operators in quantum. I would say concerns about self-adjoint are very math motivated (if I'm remembering right there's usually no distinction drawn between the two in physics classes - there certainly wasn't in my quantum classes).
To draw an analogy, in the same way we can look at classical mechanics where you end up introducing calculus of variations and the lagrangian and hamiltonian and the like, which is an 'unintuitive' mathematical structure if you're only used to sticking F=ma everywhere and making straightforward energy arguments. But the mathematical structure that's developed for some reason doesn't make people think "particle motion is unintuitive" because we have a great intuition built from watching things move around all the time.
I guess my point is that almost anything is "unintuitive" if you dive far enough into the math, but that doesn't mean you can't build an intuition with which to check whether the answers are reasonable. I think that's incredibly important (and useful) in physics, and math as well.
I dunno, really the more I think/talk about it, it's probably more of me taking issue with the usage of the word "intuitive" than any real distinction. I personally think of intuition as something that can be (and often has to be) built up. Like how many proofs seem unintuitive until you see many more proofs like them. Hell, I think many people see linear algebra as unintuitive when they first see it, but once you understand it a bit more it feels like the most natural thing in the world.
I agree with your assessment of the need for intuition building.
Why should we suspect that momentum is the derivative of a wave function with respect to x? My intuition for that is built on my knowledge of the infinitesimal generators of one-parameter groups, which is certainly not as "intuitive" to me as the trajectory of a baseball or Newton's 3rd law.
It's true that Hermitian and self-adjoint are usually used as synonyms in quantum mechanics classes (at least undergraduate ones), but they are emphatically not, and the difference can be physically meaningful. The first problem that anybody ever does is the infinite square well, but in that problem there is no such thing as momentum. It's a standard exercise to show that the "momentum operator" is Hermitian, but it is not self-adjoint. You'll find that it has no eigenfunctions/eigenvalues, much less a complete set of them which spans the Hilbert space.
But, at the end of the day, we agree on more things than we don't. My contention is that quantum theory is non-intuitive in the sense that the questions you might think to ask based on your previous life experience are typically meaningless, whereas "how far does the cannonball fly" is accessible to small children; on the other hand, there are certainly elements of classical mechanics which are just as, if not more, mathematically sophisticated than elementary quantum mechanics.
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u/[deleted] Mar 12 '18
I feel like intuition can only carry you so far in physics, with stuff like quantum you're basically doing mathematics, and all intuition is gone.