Hi, I encountered the following HW problem which really confuses me. Could anyone please explain it to me? Thank you so much! The result of applying a Hermitian operator B to a normalized vector |1> is generally of the form: B|1> = b|1> + c|2> where b and c are numerical coefficients and |2> is a normalized vector orthogonal to |1>. My question is: Why B|1> must have the above form? Does it mean if |1> is an eigenstate of B, then b=!0 and c=0? But what if |1> is not an eigenstate of B? I also need to find the expectation value of B (<1|B|1>), but I think I got this part: <1|B|1> = <1|b|1> + <1|c|2> = b<1|1> + c<1|2> = b since |1> and |2> are orthogonal and they're both normalized. Does that look right?