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Is it customary to speak of the volume of some region of Hilbert space? I have no doubt one can concoct measures that make it possible, but I'm asking if it's done in practice. I'm particularly thinking of the notion that quantum states correspond to vectors on the unit sphere in Hilbert space. If n>7 or so, the volume of the unit n-ball decreases as n increases, with a limit of 0 as n approaches infinity. So would we say the unit sphere in Hilbert space has zero volume? What about the unit cube?
I thought about the unit sphere because if it had positive volume, one could say from the curse of dimensionality that the volume was entirely concentrated at the surface, so in some sense quantum states as unit vectors couldn't be distinguished from random points in the unit ball, and I wondered if that could mean anything physically. I don't actually know any QM so am trying to make some sense of it by understanding little aspects like this. Thanks.
173.228.123.121 ( talk) 11:17, 5 March 2018 (UTC)
Our article Root of unity#Algebraic expression says, concerning the sixth-degree cyclotomic polynomial giving the primitive 7th roots of unity,
Thanks in advance, Loraof ( talk) 16:41, 5 March 2018 (UTC)
Mathematics desk | ||
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< March 4 | << Feb | March | Apr >> | March 6 > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
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The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
Is it customary to speak of the volume of some region of Hilbert space? I have no doubt one can concoct measures that make it possible, but I'm asking if it's done in practice. I'm particularly thinking of the notion that quantum states correspond to vectors on the unit sphere in Hilbert space. If n>7 or so, the volume of the unit n-ball decreases as n increases, with a limit of 0 as n approaches infinity. So would we say the unit sphere in Hilbert space has zero volume? What about the unit cube?
I thought about the unit sphere because if it had positive volume, one could say from the curse of dimensionality that the volume was entirely concentrated at the surface, so in some sense quantum states as unit vectors couldn't be distinguished from random points in the unit ball, and I wondered if that could mean anything physically. I don't actually know any QM so am trying to make some sense of it by understanding little aspects like this. Thanks.
173.228.123.121 ( talk) 11:17, 5 March 2018 (UTC)
Our article Root of unity#Algebraic expression says, concerning the sixth-degree cyclotomic polynomial giving the primitive 7th roots of unity,
Thanks in advance, Loraof ( talk) 16:41, 5 March 2018 (UTC)