I am having problem to see the contradiction of the following working:
Consider two arbitrary operators $\hat A$, $\hat B$, such that $[\hat A, \hat B ] = c\hat1$, where $c$ is a non-zero constant. If we further assume $\hat A$ is hermitian and obey the eigenvalue equation given by $$A\left|a\right> = \lambda_a\left|a\right>,$$ where $\lambda_a$ is real number, consider the expression $\left<a\right|[\hat A, \hat B ]\left|a\right>$:
On the one hand, $\left<a\right|[\hat A, \hat B ]\left|a\right> = \left<a\right|c\hat1\left|a\right> = c\left<a|a\right> \neq 0$,
but if we expand the commutator, $\left<a\right|[\hat A, \hat B ]\left|a\right> = \left<a\right|\hat A \hat B - \hat B \hat A\left|a\right> = (\left<a\right|\hat A) \hat B \left|a\right> - \left<a\right|\hat B (\hat A\left|a\right>) = \lambda_a\left<a|B|a\right> - \lambda_a\left<a|B|a\right> = 0$.
I could not see why these two results are different, I assume there's some properties of the associativity with respect to the commutator I am missing, but I can not figure it out formally.
