How Does Spin Interaction with a Magnetic Field Affect Quantum States?

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SUMMARY

The discussion centers on the interaction of spin states with a magnetic field, specifically addressing a problem from quantum mechanics involving the Hamiltonian operator Bo. The participant attempts to solve for the quantum state |ψ(t)> but encounters issues with obtaining a column vector solution, which is essential for calculating the probability using the inner product ||². The confusion arises from the relationship between the spin operator S_y and the Hamiltonian H, both of which are represented as matrices in this context.

PREREQUISITES
  • Understanding of quantum mechanics, particularly spin states and operators
  • Familiarity with Hamiltonian mechanics and its application in quantum systems
  • Knowledge of matrix representation of quantum operators
  • Ability to compute inner products in a quantum context
NEXT STEPS
  • Study the role of the Hamiltonian in quantum mechanics, focusing on specific examples
  • Learn about the matrix representation of spin operators in quantum mechanics
  • Explore the implications of magnetic fields on quantum states, particularly in the z direction
  • Investigate the calculation of probabilities using inner products in quantum mechanics
USEFUL FOR

Students and researchers in quantum mechanics, particularly those focusing on spin interactions and Hamiltonian dynamics, will benefit from this discussion.

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Homework Statement


The problem is stated on page 10 and it is number 9. http://panda.unm.edu/pandaweb/graduate/prelims/QM_F11.pdf

Homework Equations


I know how to write Sy in the z basis for either spin up or down and I have all the spinors.

The Attempt at a Solution


I tried solving for |ψ(t)> by using Bo as the hamiltonian but that seems to give me an answer that a is single solution and I was expecting a column vector as a solution. If I got a column vector I would just take the inner product of |<Sy|ψ(t)>|2 and that would give me the probability. With this said I feel like I am approaching this problem wrong and I have to take into account that fact that the magnetic field is in the z direction.
 
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Hmm what you say above doesn't make much sense to me because S_y and H are supposed to be matrices.

:confused:
 

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