Graduate Dirac Notation: Why is order reversed in ket expasion?

[RoadDog]

TL;DR

When expanding a ket as a sum of components and basis unit vectors, Why is the order of ket and corresponding vector component reversed when writing the vector component as an inner product under the summation?

Shankar Prin. of QM 2nd Ed (and others) introduce the inner product:

<i|V> = v_i ...(Shankar 1.3.4)

They expand the ket |V> as:

|V> = Σ v_i|i>

|V> = Σ |i><i|V> ...(Shankar 1.3.5)

Why do they reverse the order of the component v_i and the ket |i> when they write the former as the inner product <i|V>? It should not matter right? The reversal of order is almost as if it is to stress the appearance of the outer product |i><i|.

 

[hutchphd]

Operationally one must be careful because the operators do not commute. This leads to complications better explained by those fluently conversant in Hilbert.

 

[RoadDog]

Thank you for your reply. You are speaking of the outer product as the projection operator? Right. But I am asking, why must it be written as such. What is wrong with writing:

|V> = Σ <i|V> |i>

if <i|V> = v_i?

 

[hutchphd]

Both $v_i$ and $\bra i \ket v$ are numbers so their position does not really matter. It is a convention (a useful one) to leave the open operators on the outside.

 

[RoadDog]

OK thanks that is what I figured. Thanks