Relativistic quantum mechanics

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SUMMARY

The discussion centers on the relationship between the Dirac equation and the Schrödinger equation within the framework of relativistic quantum mechanics. It is established that the Dirac equation requires a mass term to describe particles with mass, while the massless Dirac equation is known as the Weyl equation. The Schrödinger equation, although formulated in Minkowski 4-space, is not Lorentz invariant and thus does not qualify as a relativistic equation. The Dirac equation, on the other hand, is shown to describe a 4-vector, making it inherently more suitable for relativistic applications.

PREREQUISITES
  • Minkowski metric and Lorentz transformations
  • Dirac equation and its mass term implications
  • Schrödinger equation and its limitations in relativistic contexts
  • Understanding of 4-vectors and their role in quantum mechanics
NEXT STEPS
  • Study the implications of the mass term in the Dirac equation
  • Explore the Weyl equation and its applications in particle physics
  • Learn about Lorentz invariance and its significance in quantum field theory
  • Investigate the derivation of the Schrödinger equation from the Dirac equation in the non-relativistic limit
USEFUL FOR

Physicists, quantum mechanics students, and researchers interested in the foundations of relativistic quantum mechanics and the interplay between different quantum equations.

  • #31
For example, the Dirac equations should be of the form:
$$(\partial_0 - \partial_1 + i\partial_2)\psi_1 + \partial_3 \psi_2 = - im\psi_4$$
Plus three more similar equations. There's no immediate way to decouple these: they are fundamentally equations for a four-component spinor. Certainly the SDE is nowhere to be seen.
 
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  • #32
Sure you get the non-relativistic limit by a formal expansion with respect to powers of ##1/c##, and also choosing a convenient representations for the Dirac matrices to do that. E.g., if you minimally couple the em. field you are lead to the Pauli equation with a Pauli spinor (throughing away the anti particles). All this can be found in Bjorken&Drell.
 
  • #33
redtree said:
the Schrödinger equation is a component of the Dirac equation

No, it isn't, it's a non-relativistic approximation to it, as has already been pointed out.

The OP question has been answered. Thread closed.
 

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