Sakurai Modern Quantum Mechanics (Second Edition) Eq. 1.7.15

[classy cal]

Homework Statement

Reading Sakurai I cannot see how he gets to the end of 1.7.15 as below:

Homework Equations

∫dx'|x'><x'-dx' |α>
= ∫dx'|x'>{<x' |α>-Δx'∂/∂x'<x' |α>}

The Attempt at a Solution

I tried a Taylor expansion but cannot see how this is derived.

 

[blue_leaf77]

That's indeed a Taylor expansion, truncated up to the first order. The author wants to express the shifted position eigenket as in terms of infinitesimal change.

 

[classy cal]

Thank you for your reply, it has helped me understand manipulation of bra-kets.