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Hi, suppose that the operators $$\hat{A}$$ and $$\hat{B}$$ are Hermitean operators which do not commute corresponding to observables. Suppose further that $$\left|A\right>$$ is an Eigenstate of $$A$$ with eigenvalue a.
Therefore, isn't the expectation value of the commutator in the eigenstate:
$$\left<A\right|\left[\hat{A},\hat{B}\right]\left|A\right>=\left<A\right|\left(\hat{A}\hat{B}-\hat{B}\hat{A}\right)\left|A\right>$$
Now if I act with the operator A on the two sides onto the Eigenstates, I would get:
$$=a\left<A\right|\hat{B}\left|A\right>-a\left<A\right|\hat{B}\left|A\right>=0$$
Obviously I went wrong somewhere because the operators do not commute (and, for example for x and p, the commutator is just a number...which obviously can't ever have expectation value 0). But for the life of me I can't figure out why... and this is really bothering me now. So help please?
Therefore, isn't the expectation value of the commutator in the eigenstate:
$$\left<A\right|\left[\hat{A},\hat{B}\right]\left|A\right>=\left<A\right|\left(\hat{A}\hat{B}-\hat{B}\hat{A}\right)\left|A\right>$$
Now if I act with the operator A on the two sides onto the Eigenstates, I would get:
$$=a\left<A\right|\hat{B}\left|A\right>-a\left<A\right|\hat{B}\left|A\right>=0$$
Obviously I went wrong somewhere because the operators do not commute (and, for example for x and p, the commutator is just a number...which obviously can't ever have expectation value 0). But for the life of me I can't figure out why... and this is really bothering me now. So help please?
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