Hermitian operator represented as a unitary operator

[nomadreid]

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.

 

[DEvens]

" ... 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 )^#?

 

[nomadreid]

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^†