# Homework Help: Ket Notation - Effects of the Projection Operator

1. Jan 31, 2012

### Questioneer

Ket Notation -- Effects of the Projection Operator

1. The problem statement, all variables and given/known data
From Sakurai's Modern Quantum Mechanics (Revised Edition), it is just deriving equation 1.3.12.

2. Relevant equations
$$\begin{eqnarray*}\langle \alpha |\cdot (\sum_{a'}^N |a'\rangle \langle a'|) \cdot|\alpha \rangle \end{eqnarray*}$$
3. 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?

2. Jan 31, 2012

### fzero

Re: Ket Notation -- Effects of the Projection Operator

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.

3. Jan 31, 2012

### Questioneer

Re: Ket Notation -- Effects of the Projection Operator

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?

4. Jan 31, 2012

### fzero

Re: Ket Notation -- Effects of the Projection Operator

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.