Sakurai Modern Quantum Mechanics (Second Edition) Eq. 1.7.15

Click For Summary
SUMMARY

The discussion focuses on the derivation of Equation 1.7.15 from Sakurai's "Modern Quantum Mechanics" (Second Edition). The equation involves the manipulation of position eigenkets and the application of a Taylor expansion to express the shifted position eigenket in terms of an infinitesimal change. The key takeaway is that the expression utilizes a first-order Taylor expansion to achieve the desired result, clarifying the relationship between the bra-ket notation and infinitesimal variations.

PREREQUISITES
  • Understanding of quantum mechanics concepts, specifically bra-ket notation.
  • Familiarity with Taylor series expansions and their applications in physics.
  • Knowledge of operator manipulation in quantum mechanics.
  • Basic grasp of infinitesimal calculus and its relevance in quantum theory.
NEXT STEPS
  • Study the application of Taylor expansions in quantum mechanics, focusing on position eigenkets.
  • Explore the manipulation of bra-ket notation in quantum mechanics for deeper insights.
  • Review Sakurai's "Modern Quantum Mechanics" for additional context on operator theory.
  • Investigate the implications of infinitesimal changes in quantum state representations.
USEFUL FOR

Students of quantum mechanics, physicists working with quantum state representations, and anyone seeking to deepen their understanding of operator manipulation and Taylor expansions in quantum theory.

classy cal
Messages
7
Reaction score
0

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.
 
Physics news on Phys.org
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.
 
Thank you for your reply, it has helped me understand manipulation of bra-kets.
 

Similar threads

Quantum Modern Quantum Mechanics 3rd Ed: J. J. Sakurai & Jim Napolitano Review
  • · Replies 10 ·
Replies
10
Views
7K
Writing a squared observable in Dirac notation
  • · Replies 9 ·
Replies
9
Views
2K
Studying Struggling with exercises after reading quantum mechanics textbook
  • · Replies 2 ·
Replies
2
Views
1K
What is the Momentum Operator Identity in Modern Quantum Mechanics?
  • · Replies 6 ·
Replies
6
Views
4K
Ket Notation - Effects of the Projection Operator
  • · Replies 3 ·
Replies
3
Views
3K
Quantum Video lecture suggestion for Sakurai quantum mechanics Textbook
  • · Replies 3 ·
Replies
3
Views
4K
What is the role of linear operators in quantum mechanics?
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Sakurai page 54: Is this a Taylor expansion?
  • · Replies 7 ·
Replies
7
Views
2K
Operators in Quantum mechanics: can one swap \Psi and \Psi^*
  • · Replies 9 ·
Replies
9
Views
2K
Quantum "Modern Quantum Mechanics" by Sakurai for First Encounter
  • · Replies 4 ·
Replies
4
Views
4K