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I'm looking for counterexamples in quantum theory, in the spirit of books like Counterexamples in topology and Counterexamples in analysis. A practically identical post, but for PDEs, can be found here.

This request will probably be clearer to people who have read those books: I am not looking for claimed 'disproofs' of quantum theory by cranks, but rather examples from quantum theory that defy intuition or expectation, or which 'break' theorems that we would otherwise want to hold, and in the process help clarify some of the theory's more subtle issues. Also note that I am not looking for ways in which quantum theory deviates from classical physics, like the tunneling effect or the blackbody spectrum.

A good example might be a strange solution to the Schrodinger equation, or to the Dirac equation in a particular field, or maybe something like the Haag–Lopuszanski–Sohnius theorem, etc.

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closed as not constructive by David Z Mar 25 '13 at 7:06

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To be clear: are you asking for mathematical constructions in the framework of quantum theory (i.e. the $1/r^2$ potential which has some bizarre properties), or physically realized examples? +1 for the idea though - love the counterexamples series! –  Michael Brown Mar 25 '13 at 6:38
    
Either one would be fine. I think a good "physically realized example" from, say, optics, might be the fact (only recently discovered) that you can have fractal laser modes. I'm not sure what that would be a counterexample to, though, but it's the sort of head-turning "seriously? that's possible?" thing I'm after. –  Andrew Gibson Mar 25 '13 at 6:43
    
I'd like to hear about the 1/r^2 potential. Maybe you could elaborate in an answer. –  Andrew Gibson Mar 25 '13 at 6:50
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Hi Andrew - unfortunately list questions like this are not appropriate for this site. You could try to collect some ideas in Physics Chat though. –  David Z Mar 25 '13 at 7:06
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@AndrewGibson check this book out: amazon.com/Quantum-Paradoxes-Physics-Textbook-Aharonov/dp/… –  Mark Mitchison Mar 25 '13 at 10:27