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The three possibilities that you talked about are very different from each other... To do research about black holes in high school is pretty much impossible because you need to have background in GR and QFT. I don't know what you mean exactly by "doing research about wave/particle duality". You have to be more specific. Same comment for Newton's Laws of ...

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There are some very good physics problems involved in counting modes. Actually, the cases where you have to actually count the modes are rather advanced, because you come up against the high-frequency cutoff situations and that involves some heavy-duty physics. But listing the modes is a good math problem. To do this with a high-school class you'd have to ...

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I think the angle $2 \theta$ is used because we can see diffracted pattern from incident beam so the angle of incident and reflected are combine to become $2 \theta$.

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The displacement of 7.44° is clearly wrong. It is inconceivable that a torsion pendulum with a period of around 50 - 100 seconds could be displaced by such a large amount through the attraction of a couple of 2 kg masses. I have to conclude that other factors (air currents?), not gravity, were the cause of the displacement you observed. You really need to ...

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I think Shankar's Principles of Quantum Mechanics contains decent exercises that promote understanding of the material, many of which are proofs and derivations rather than simply computations.

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The weird symbols that you see in the textbooks are notations used in the theory of linear operators on Hilbert spaces, which is the foremost prerequisite to approach classes in quantum mechanics. In a nutshell, it is the infinite-dimensional version of linear algebra. Notice that infinite dimensions do bring different consequences (almost all that quantum ...

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IMO, if you read this book, it will give you a pretty good idea of what you need to learn in order to learn modern physics. (Though, it does put more emphasis on GR than on QM). http://www.amazon.com/Road-Reality-Complete-Guide-Universe/dp/0679776311/

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The answer depends on two things: 1. What is your current baqckground, and 2. What is your purpose in learning quantum mechanics. The background in physics that you need is classical mechanics, of course. This is minimum. But something about heat and electricity is also very desirable. In mathematics you should be familiar with Calculus in one and several ...

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Quantum interactions behave in a much more wave-like way than a pinball model would suggest. The interactions are quantised, which is where a lot of the particle-like behaviour comes from, but until an interaction depends on the specific position of the electron, it will continue to behave much more like a wavy cloud than a ball. The pinball will also fail ...

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In x-ray powder diffraction, you have crystallites in all possible orientations. Only those crystallites whose bragg planes are at an angle θ with respect to the incident angle will diffract at an angle 2θ with respect to the incident beam (or at an angle θ with respect to the diffracting planes). So that is the reason, you always use 2θ instead of θ.

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