|MadSci Network: Physics|
Hi, Jeremy. You’re asking a very interesting question – one that goes to the heart of quantum mechanics. To begin with, let’s define what ‘conjugate attributes’ means. First, I’m going to refer to ‘conjugate variables’ rather than attributes. (A Google search under conjugate attributes turns up philosophy sites, rather than physics sites.) The generally-accepted definition that I’m going to choose can be seen, for example, at http://www.campusprogram.com/reference/en/wikipedia/uncertainty_ principle.html and states that two variables are conjugates if their quantum operators have a non-zero commutation relation. That’s it – all variables with non- commuting operators are conjugate variables. Because there are so many (an infinite number, actually) I won’t go into even a partial list here. For any operator that you can think of, angular momentum, spin, intensity, etc., any other operator that doesn’t commute with it forms a conjugate pair. I imagine that you associate conjugate variables with the uncertainty principle. There’s a good reason for that: all conjugate variables have an associated uncertainty relation, with the minimum uncertainty related to their commutator. See the link at ht tp://scienceworld.wolfram.com/physics/UncertaintyPrinciple.html for the proof. There’s a technicality concerning the uncertainty relation between E and t, since t is not a variable with an associated operator. I leave that to the links outlined at the end of this answer. Lastly, if I understand your question, “Oh and also if you could tell me what wave form represents the attribute” correctly, then you’re referring to an eigenstate of the corresponding operator. For example, the eigenstate of the position operator is a delta function, while the eigenstate of the momentum operator is a plane wave. However, I feel compelled to point out that both of these are just the spatial representations of these wavefunctions, and the power of the wavefunction is that it isn’t confined to just one set of coordinates. You can equally-well describe a wavefunction in momentum space, in which case the eigenstate of the momentum operator is a delta function and that of the position operator is a plane wave. It’s the same thing, the same eigenstate representing the same information, but viewed from a different set of coordinates. Good luck, and keep asking questions! Good intro QM book: ================ My favorite: “Principles of Quantum Mechanics,” R. Shankar ISBN 0-306- 40397-8 Technicality concerning the ‘uncertainty’ relationship between E and t: =================================================== http://math.ucr.edu/h ome/baez/uncertainty.html Also D. Shalitin, “On the time-energy uncertainty relation,” American Journal of Physics, vol 52, iss 12, p 1111, 1984, link (for those w/ access to library subscriptions) at: http://scitat ion.aip.org/getpdf/servlet/GetPDFServlet? filetype=pdf&id=AJPIAS000052000012001111000001&idtype=cvips
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