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A common misunderstanding of quantum mechanics is the belief that EVERYTHING in the world is quantized, but this is simply not true. For example the position of a free particle is not quantized but may take on any value.

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You can resolve the E vectors coming out of the yellow surface into a component parallel to the surface and a component perpendicular to the surface: $E = E_{parallel} + E_{perpendicular}$. As I say to my students a lot of the time, draw the triangle! The triangle here has a hypotenuse of one of the E vectors (in red), one leg of the triangle is the ...

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Small point to add to this, but all matter has a Schwarzschild radius - the earth does (it's about 1/3rd of an inch in diameter), black holes do - it's their event horizon, and Neutron Stars do - but it's smaller than the star. Any non-black hole object has a Schwarzschild radius that is smaller than it is. Neutron Stars are dense enough that their ...

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Comments to the question (v3): The possible quantum states depend on the topology of spacetime. For instance, the momentum in the extra (fifth) direction of a 5-dimensional Kaluza-Klein theory is quantized/discrete variable if the extra (fifth) dimension is a compact circle $S^1$, but a continuous variable if the extra (fifth) dimension instead is a ...

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OP comments as an example of what the question is about: Let us consider the case of an electron confined to a curved surface. Does the geometry of the background have any consequences for the state space? A simple answer can be given for ordinary QM: A (scalar, i.e. spin-0) particle moving in one-dimension has state space $L^2(\mathbb{R})$, a particle ...

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Please correct me if I misunderstood. As I understand your question, you have a bike lying on its side. You grab the steering handles and lift up until the bike is standing normally on the tires ready for you to mount it. So, I will assume that the tires are touching the ground throughout this lifting, right? So you are never lifting the "whole" bike off ...

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