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I have some trouble to prove the next statements:

  1. Let $A,B$ two anti-commutative hermitian operators, i.e. $\{A,B\}=AB+BA=0$. Does $A$ and $B$ share any eigenket?.

  2. If $U$ is an unitary operator such that $U=A+iB$ with $A$ and $B$ hermitian operators with non-degenerated eigenvalues. Show that any eigenvector of $A$ is an eigenvector of $B$.

For $1.$ "Let $v$ a common eigenket, thus $Av=av$ and $Bv=bv$ so:

$0=(AB+BA)v=ABv+BAv=(ab+ba)v$, since $A,B$ are hermitian and $a,b$ are real, and $ab=-ba=-ab$ thus $ab=0$ but I don't know if this tells me something, and for $2.$ I don't know how to start, Any help?

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    $\begingroup$ According to your derivation for (1), it said if $A$ and $B$ share a common eigenket, then $a = 0$ or $b = 0$. Equivalent, it said if both $a$ and $b$ are non-zero, then $A$ and $B$ does not share any eigenket $\endgroup$ – K_inverse Sep 24 '18 at 7:17

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