- #1

nomadreid

Gold Member

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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.