Berry phase, Bra-Ket and gradient

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SUMMARY

The discussion focuses on deriving an equation involving the curl of a vector field expressed in terms of gradient and ket-bra notation. Participants emphasize the need for clarity in notation and context, particularly regarding the cross product between ket and bra vectors. The parameter space R is defined as a three-dimensional space, simplifying the application of vector algebra. The equation in question is referenced from a specific source, indicating a need for further exploration of its derivation.

PREREQUISITES
  • Understanding of vector calculus, specifically curl and gradient operations.
  • Familiarity with quantum mechanics concepts, particularly ket and bra notation.
  • Basic knowledge of eigenfunctions and their role in physical systems.
  • Ability to interpret mathematical notation in the context of physics.
NEXT STEPS
  • Study vector calculus focusing on curl and gradient operations in three-dimensional space.
  • Learn about ket and bra notation in quantum mechanics and their mathematical implications.
  • Explore the derivation of equations involving eigenfunctions in quantum systems.
  • Review the referenced book for context on the equation and its applications in physics.
USEFUL FOR

Students and professionals in physics, particularly those studying quantum mechanics, as well as mathematicians interested in vector calculus and its applications in physical theories.

Lucy166
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Could somebody show me how to derive this equation? How can I get right side from left. Step by step, thanks...
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I've no clue, what your notation means. We need more context!
 
I only want to know how to solve this curl in left side when I have gradient and ket-bra vectors. I suppose it is elementary vector algebra. It’s simply a math. I need to show that solution of this curl is right side of this equation in this form.

R (external variables) is set of parameters (R1, R2, R3…). In this section, for sake of simplicity, the parameter space R is assumed to span an ordinary three dimensional space. We don’t need generalization of vector algebra to a multi-dimensional space. ∂/∂R=∇R. We consider eigenfunctions psi. Internal variables of the system are collectively indicated by r.
 
Then it's even more puzzling. What should a cross product between a ket and a bra mean? Again, we need the context and a clear definition of your notation!
 
Lucy166 said:
this equation?

Where does this equation come from?
 

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