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Hermitian operator represented as a unitary operator

  • Thread starter nomadreid
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  • #1
nomadreid
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Homework Statement


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.

Homework Equations


U=exp(iK)
Letting M# be the adjoint of M (I can't find a dagger), and * be multiplication,
unitary: M#*M = M*M# = I (Identity operator)
Hermitian:M#=M

The Attempt at a Solution


Given a Hermitian operator K, define U as above, and then one must prove that exp(iK)*(exp(iK))# = I. That is, that (exp(iK))# =exp(-iK)
Obviously I must use the fact that K=K# to do so, but this only makes the problem to show that exp(-iK#) = (exp(iK))# , which doesn't seem to get me very far. The fact that (M#)-1=(M-1)# 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.
 

Answers and Replies

  • #2
DEvens
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" ... exp(-iK#) = (exp(iK))# ..."

Can you show that? Can you do some operations on exp(iK) that will allow you to bring the # operator into the ()? It's hard to know how much of a hint is considered acceptable in a homework forum, but how about Taylor? So, given that K# = K, what is ( i K * K )#?
 
  • #3
nomadreid
Gold Member
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Thanks, DEvens. Let me see if I see where your hints are leading. First, since if K is Hermitian so is iK, so
(iK2)#=(iK*K)#=(K#(iK)#)=-i(K#)2.
Similarly
(iK3)#=(iK2*K)#=(iK2)#*K#=
=(-i(K#)2)K#=-i(K#)3
and so forth, so
expanding (exp(iK))# via Taylor, we get (∑ (iK)n/n!)# = (∑ (-iK#)n/n! = exp(-iK#)
Is this correct?
PS I found the dagger M
 

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