1
$\begingroup$

I have a Hermitian unitary Ĥ and I want to know, if Û is some other unitary, when is Ĥ Û a Hermitian unitary? Specifically, what are the conditions on Û such that $(\hat{HU})^{\dagger}=\hat{HU}$?

I know that if Û is Hermitian then as long as they commute this is true. Also that if they commute, as long as Û is Hermitian this is also true.

So then does it have to be the case that they commute AND Û is Hermitian?

Thanks a lot for the help in advance!

$\endgroup$
1
$\begingroup$

$(HU)^* = U^* H^* = U^{-1} H$, so what you need is $HU = U^{-1} H$, i.e. $U H U = H$. $H$ and $U$ do not need to commute. For example, try the matrices

$$ H = \pmatrix{1 & 0\cr 0 & -1\cr}, \ U = \pmatrix{\cos(t) & -\sin(t)\cr \sin(t) & \cos(t)\cr}$$ where $\sin(t) \ne 0$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.