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Study Quantum Physics

I am a high schooler who is interested in physics and mathematics, and I have a kind of 'high-school thesis' coming up in a year and a half or so. I want mine to be about Quantum Physics, and I have already prepared by self-studying Linear Algebra (and I'm planning on starting with Differential Equations too). I just have a couple of obstacles:

  • If I know Linear Algebra and (Ordinary & Partial) Differential Equations, what parts of Quantum Physics will be in my reach?

  • What other fields of mathematics would you recommend me studying to gain a grasp of rudimentary quantum physics and to make me able to make a thesis on quantum physics?

  • What would you guys consider interesting for a thesis on quantum physics? What are fascinating experiments, concepts, results, etc. of quantum physics that would be worth making a thesis about?

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related: physics.stackexchange.com/questions/38963/… –  DJBunk Oct 26 '12 at 16:06
    
Are you looking for an experiment or a study? If you could reproduce the MIT double-slit in a box I'd give you an A+ myself :) physics.stackexchange.com/questions/16783/… –  AlanSE Oct 26 '12 at 18:04
    
Related questions by OP: physics.stackexchange.com/q/39165/2451 and physics.stackexchange.com/q/38963/2451 –  Qmechanic Oct 26 '12 at 19:41
    
@Qmechanic How on earth is this a duplicate? Thank you for closing this question. This site is for spreading knowledge, not preventing it. –  kamal Oct 27 '12 at 12:04
    
Dear @kamal: You have now asked 3 related soft-questions about how to study quantum physics. This site is more suited for actual physics questions. –  Qmechanic Oct 27 '12 at 13:14
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marked as duplicate by Qmechanic, David Z Oct 27 '12 at 8:59

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I think most people would agree that Bell's theorem is one of the most fascinating and perplexing results in quantum theory. In short, Bell's theorem quantifies the non-separability of quantum mechanics in precise terms. Non-separability means that, contrary to intuition, it is sometimes not possible to talk about physical properties (such as momentum, position etc.) belonging to one particle alone.

Even if two electrons are a million light-years apart, if they are in an entangled state (which will generally be true if they interacted in the past) then the outcomes of measurements on the two particles are correlated. Bell showed that these correlations are such that the measurement outcomes cannot be explained by properties belonging to one particle or the other, rather you can only speak about properties belonging to the system as a whole (in this case the system means the pair of electrons together). This strikes me (and a lot of other people too, notably Einstein) as pretty weird, since you wouldn't expect the momentum of electron A that is sitting here in your lab on Earth to depend on electron B, which could be a million light years away!

Alternatively, Bell's theorem shows that if you want to explain the outcomes of your measurements in terms of properties belonging just to one electron or the other, then the electrons must be able to communicate instantaneously over arbitrarily large distances, which does not sit well with our understanding of other laws of physics like relativity. The amazing thing is that Bell's theoretical result was proved correct by experiments by Alain Aspect in the 80's (if I remember right).

Hopefully that convinces you that Bell's theorem is interesting. It is a suitable topic for your project because the maths required is minimal: you already know more than enough to understand it (although perhaps learning a bit of elementary probability theory would help in reading Bell's original papers). That does not mean that the physics won't make your head hurt though :) There are also lots of other related things you could talk about: the experiments that confirmed Bell's theorem, the relationship of entanglement to quantum computing etc.

You can find out more by reading Bell's book "Speakable and Unspeakable in Quantum Mechanics", which anyway should be required reading for any budding student of quantum mechanics.

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High school is age 14-18 isn't it? Studying Bell's theorem at that age is going to be a challenge. –  John Rennie Oct 26 '12 at 16:07
    
I disagree. The actual maths used in the proof is trivial: you only need an elementary understanding of linear algebra and perhaps a bit of probability theory (will edit to include this). And the relevant physical concept under investigation, locality, is intuitively understood by any 5 year old. There are also more popular/philosophical accounts (beautiful explanation in Lange for example: amazon.com/An-Introduction-Philosophy-Physics-Locality/dp/…) for someone who wants to understand the physical content of Bell's theorem without being able to repeat the derivation. –  Mark Mitchison Oct 26 '12 at 16:14
    
The math of quantum mechanics is easy, its always linear, the math of classical is leagues harder since there is no linearity requirement. Math isn't the hard part of quantum, its the physics IMO. –  kηives Oct 26 '12 at 16:18
    
@knives I couldn't agree more. My comment still stands though: the physical concept of locality is a deep-rooted intuition. This makes the content of Bell's theorem much easier to understand (and obviously much harder to accept) than most other interesting topics in quantum physics, which typically either involve quantum many-body systems or are so esoteric that I doubt a high-school physics teacher would approve of them. –  Mark Mitchison Oct 26 '12 at 16:24
    
@MarkMitchison Wow, this is funny and a bummer at the same time. Last year, a boy at our school did Bell's theorem and won a national prize with it. His parents are both experimental physicists and he was interested in physics too. So I probably can't do it too, it would look like I'm plagiarizing him. –  kamal Oct 26 '12 at 18:29
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Here is a nice thing you can do at a high school level--- show if it is possible to take a photograph in complete darkness.

Suppose you have a dark room, and you don't want to disturb it, you don't want to shine light on it. But you want to see what's in the room. You have access to a light source, a window into the room (or you are in the room), beam-splitters, and interferometers, but you want to ensure that at any time the probability of a photon getting detected inside the room is negligible. Can you take a picture?

Classically you can't do it, you either shine a photon into the room, or you don't. But, with a modification of the Elitzur-Vaidman bomb tester, you should be able to easily build a theoretical device which can scan a photograph of the room without ever allowing an appreciable probability of a photon ever being detected inside the room, outside your apparatus.

The mechanism has not been worked out in full, this is an old original idea of mine, but it is simple given the Elitzur Vaidman result. A sketch is as follows--- you do the Elitzur Vaidman thing, split a photon into N components with magnitude $\epsilon$, where $N\epsilon^2$ is small, and scatter all these little photon components off the room in separated bunches. You then have a rotating lens which collects the photon amplitude fraction that come back out the window, and interferes with the major component of the beam. You can detect interference when $N\epsilon$ is non-negligible, but this still makes it that a person inside the room will detect no photons. You can make the illumination going into the room arbitraily small, while allowing the interference effects to be detectable, and to reveal at which angles the photon returns to you, or if it is absorbed.

I do not know if this can be used to make a practical dark camera, but it is possible in principle, and it would make a nice high-school project which I think would be guaranteed a top prize, even if it is purely theoretical. It's another counterintuitive property of quantum mechanics, the counterfactual measurement.

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Well, I found out that, if you do a project on any of the sciences, it MUST be technical. I originally wanted to make my thesis on the constants of $c$ and $\hbar$, and what would happen if $c$ was smaller and $\hbar$ was larger, and by doing that explaining some QM and relativity.. however, it is purely theoretical .. –  kamal Oct 27 '12 at 12:38
    
Otherwise this sounds extremely interesting, and if the dark camera can actually be made, that certainly be fantastic. –  kamal Oct 27 '12 at 12:40
    
@kamal: I don't understand what is non-technical here (are you refering to the fact that I spoke in English rather than in formulas? That was to make it understandable, and to leave the hard work of designing the camera up to you, so you can put your name on it)--- designing such a camera, even purely theoretically, is far more technically demanding than Bell tests--- it's also completely new, no one has designed it before. Regarding varying $\hbar$ and $c$, see this answer: physics.stackexchange.com/questions/21721/… . –  Ron Maimon Oct 27 '12 at 13:25
    
the idea of the constants was shut down for being too theoretical, but I thought it would be a fun way of explaining some QM and relativity. With technical, I meant you actually have to do tests and do real-life experiments (so a purely mathematical thesis, even if you explain it and link it to the real word, is forbidden). So you MUST have something 'concrete', if you understand. If I could somehow make this camera, that of course would be sensational. –  kamal Oct 27 '12 at 15:32
    
@kamal: The camera is beyond your ability to make, it is very difficult to do, I am not sure if it can be practically done at all. But making a simple version of the bomb-tester is not necessarily out of reach, if you can get access to a modern optical lab (the equipment is expensive). Regarding the constants--- it is meaningless to say $\hbar$ and $c$ change, what is meaningful is to say atoms are bigger/smaller in Planck lengths or times. –  Ron Maimon Oct 27 '12 at 15:40
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