Berry phase of 1/2 spin in slowly rotating magnetic field

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SUMMARY

The discussion focuses on the Berry phase of a 1/2 spin particle in a slowly rotating magnetic field. Participants analyze the Hamiltonian and eigenstates, emphasizing that the spatial part of the quantum state is irrelevant since the electron is fixed in space. The conversation highlights confusion regarding the application of equation (2.6.6) and the absence of a Laplacian operator in the Hamiltonian. Key insights include the necessity of evaluating the Berry phase using the Dell operator as indicated in the relevant equations.

PREREQUISITES
  • Understanding of quantum mechanics, specifically spin systems
  • Familiarity with Hamiltonians and eigenstates
  • Knowledge of Berry phase concepts
  • Proficiency in vector calculus, particularly the Dell operator
NEXT STEPS
  • Study the derivation of the Berry phase in quantum mechanics
  • Learn about the implications of Hamiltonians without Laplacian operators
  • Explore the application of the Dell operator in quantum systems
  • Investigate the behavior of spin in arbitrary magnetic fields
USEFUL FOR

Quantum physicists, graduate students in physics, and researchers studying spin dynamics in magnetic fields will benefit from this discussion.

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Homework Statement


upload_2017-9-28_14-59-23.png


Homework Equations


upload_2017-9-28_15-4-20.png


This is the way to solve when magnetic field B is arbitrary direction one.

The Attempt at a Solution


upload_2017-9-28_15-0-2.png


I got a eigenvalue of this Hamiltonian and eigenstates.
but i have no idea how to set a coordinate to value the gradient
 

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The electron is fixed in space, hence the spatial part of the quantum state is irrelevant. You only need to concentrate on what happens to the spin.
 
DrClaude said:
The electron is fixed in space, hence the spatial part of the quantum state is irrelevant. You only need to concentrate on what happens to the spin.
upload_2017-9-28_19-8-22.png


This value is always 0 , even though i derivative (wt) by theta or pi
 

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I don't understand why you are invoking eq. (2.6.6) at all. The full Hamiltonian to be considered is given in the problem, and it has no Laplacian operator.
 
DrClaude said:
I don't understand why you are invoking eq. (2.6.6) at all. The full Hamiltonian to be considered is given in the problem, and it has no Laplacian operator.

I think you don't know how to evaluate berry phase, you should look at the last line of relevant equations. There is Dell operator
 

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