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Hermitian operator represented as a unitary operator
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[QUOTE="nomadreid, post: 4959930, member: 112452"] [h2]Homework Statement [/h2] I know that any unitary operator U can be realized in terms of some Hermitian operator K (see equation in #2), and it seems to me that it should also be true that, starting from any Hermitian operator K, the operator defined from that equation exists and is unitary. [h2]Homework Equations[/h2] U=exp(iK) Letting M[SUP]#[/SUP] be the adjoint of M (I can't find a dagger), and * be multiplication, unitary: M[SUP]#[/SUP]*M = M*M[SUP]#[/SUP] = I (Identity operator) Hermitian:M[SUP]#[/SUP]=M [h2]The Attempt at a Solution[/h2] Given a Hermitian operator K, define U as above, and then one must prove that exp(iK)*(exp(iK))[SUP]#[/SUP] = I. That is, that (exp(iK))[SUP]#[/SUP] =exp(-iK) Obviously I must use the fact that K=K[SUP]#[/SUP] to do so, but this only makes the problem to show that exp(-iK[SUP]#[/SUP]) = (exp(iK))[SUP]#[/SUP] , which doesn't seem to get me very far. The fact that (M[SUP]#[/SUP])[SUP]-1[/SUP]=(M[SUP]-1[/SUP])[SUP]#[/SUP] also seems tempting to apply somewhere, but I do not see where. Oh, of course there is also the possibility that my conjecture is wrong. Side note: I presume this should go in some Homework rubrik, because although this is not a Homework problem, it seems like something that would be given for homework in an algebra course. Therefore this should probably go under a rubrik for Algebra, but there was no Algebra listing in the Homework menu. [/QUOTE]
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Hermitian operator represented as a unitary operator
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