Ket Notation - Effects of the Projection Operator

[Questioneer]

Ket Notation -- Effects of the Projection Operator

Homework Statement

From Sakurai's Modern Quantum Mechanics (Revised Edition), it is just deriving equation 1.3.12.

Homework Equations

$$\begin{eqnarray*}\langle \alpha |\cdot (\sum_{a'}^N |a'\rangle \langle a'|) \cdot|\alpha \rangle \end{eqnarray*}$$

The Attempt at a Solution

The summation can be moved to the left, so everything is being summed from a' to N, but does an alpha bra inner product with a' (or <α|a'>) does the sum of this from all a' to N equal Ʃ<a'|α>? maybe this is simple and I just can't see it?

 

[fzero]

Check (1.2.12), which is a fundamental property of the inner product. That property holds even when the bras and kets correspond to different bases.

 

[Questioneer]

Check (1.2.12), which is a fundamental property of the inner product. That property holds even when the bras and kets correspond to different bases.

So, is it in this particular case that they are equal because we are considering the eigenkets of A, a hermitian operator? Because these are the eigenkets of A, does that mean that all operators on it are real? Even though it is the operator <α| acting on it and not A?

 

[fzero]

So, is it in this particular case that they are equal because we are considering the eigenkets of A, a hermitian operator? Because these are the eigenkets of A, does that mean that all operators on it are real? Even though it is the operator <α| acting on it and not A?

There's no assumption here that $\langle \alpha | a'\rangle$ is real, just that $\langle \alpha | a'\rangle = \langle a'| \alpha\rangle^*.$ That is why the absolute value appears in the formula.