Non-diagonality of Dirac's Hamiltonian

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In summary, the conversation discusses the non-diagonal nature of Dirac's Hamiltonian and how it affects the state of a particle with definite momentum and spin over time. The spin vector operator is not a constant of motion for the Dirac Hamiltonian, but the product of the spin and momentum operators is. This information may not be explicitly stated in some textbooks, leading to incorrect assumptions.
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Mesmerized
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Hi all,

My question is why Dirac's Hamiltonian isn't diagonal? As much as I understand, the momentum of the particle and it's spin belong to the complete set of commuting variables, which define the state of the particle, and their eigenstates must also be energy eigenstates. But because Hamiltonian is non-diagonal (because gamma matrices are not diagonal), it follows that if we have a particle with definite momentum and definite spin, with time it is going to change it's state to become a particle with no definite spin.

Where am I wrong?
 
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  • #2
The spin vector operator is no constant of motion (doesn't commute with the Hamiltonian, which is time-independent) for the Dirac' Hamiltonian, but [tex] \vec{S} \cdot \vec{P} [/tex] is.
 
  • #3
thanks, don't know why it's not written in the textbook, and I made a wrong assumption
 

FAQ: Non-diagonality of Dirac's Hamiltonian

1. What is the significance of the non-diagonality of Dirac's Hamiltonian?

The non-diagonality of Dirac's Hamiltonian is significant because it allows for the inclusion of spin into the equation, providing a more accurate description of the behavior of particles with intrinsic angular momentum.

2. How does the non-diagonality of Dirac's Hamiltonian affect the energy levels of a particle?

The non-diagonality of Dirac's Hamiltonian introduces a spin-orbit coupling term, which splits the energy levels of a particle with spin into different states. This results in a more complex energy spectrum compared to the Schrödinger equation, which only accounts for the particle's spatial coordinates.

3. Can the non-diagonality of Dirac's Hamiltonian be applied to all particles?

No, the non-diagonality of Dirac's Hamiltonian is only applicable to particles with spin, such as electrons. Particles without spin, such as photons, do not have a spin-orbit coupling term and therefore do not require the use of the Dirac equation.

4. How does the non-diagonality of Dirac's Hamiltonian impact our understanding of quantum mechanics?

The inclusion of spin in Dirac's Hamiltonian allows for a more complete description of the behavior of particles at the quantum level. It also provides a theoretical basis for the existence of antimatter, which was later experimentally confirmed.

5. Are there any practical applications of the non-diagonality of Dirac's Hamiltonian?

Yes, Dirac's equation has been used to successfully predict the existence of new particles, such as the positron, and has also been applied in the development of quantum field theories and other areas of advanced physics.

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