Is the Energy of a Spin 1/2 Particle Always Equal to h/2?

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SUMMARY

The discussion centers on the energy of a spin 1/2 particle, specifically an electron, in relation to the operator Sz and the Hamiltonian H. It is established that Sz = h/2 for the up spin state, leading to the assumption that the energy of the electron in this state is also h/2. However, when applying the Hamiltonian H = [1 0; 0 -1], the energy eigenvalue for the state |1 0> is calculated as 1, creating a discrepancy. This highlights the importance of understanding the relationship between spin operators and Hamiltonians in quantum mechanics.

PREREQUISITES
  • Understanding of quantum mechanics principles, particularly spin 1/2 particles.
  • Familiarity with Hamiltonian operators and their eigenvalues.
  • Knowledge of the Schrödinger equation and its application in quantum systems.
  • Basic grasp of linear algebra, specifically matrix representation of quantum states.
NEXT STEPS
  • Study the implications of spin operators in quantum mechanics, focusing on Sz and its physical interpretations.
  • Learn about Hamiltonians in quantum systems and how to derive energy eigenvalues from them.
  • Explore the Schrödinger equation in greater depth, particularly its role in determining energy states.
  • Investigate the mathematical framework of quantum mechanics, including matrix mechanics and state vectors.
USEFUL FOR

Students and professionals in physics, particularly those specializing in quantum mechanics, as well as researchers exploring the properties of spin 1/2 particles and their energy states.

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We know Sz = h/2 [1 0 // 0 -1] for up spin.So can we say that the energy of the electron in this state is h/2 ?
(since Sz |up> = h/2 |up>)Now we consider a Hamiltonian H = [1 0 // 0 -1] which has eigen values +1, -1 and the respective states are
(1 0) and (0 1).
If we see the Schrödinger's equation it gives :
H |1 0 > = 1 * |1 0> which says energy of | 1 0 > is = 1.

We take same hamiltonian, same state, but once we are getting energy eigen value 1/2 ,next time we are getting 1.
Where I am getting it wrong ?
 
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Well that's a bit dumb question...Sz includes "1/2" .:H
 

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