Dirac Equation

Teacher : Claude Leroy

After a brief reminder of basic mathematical tools (Lagrangian, impulse, Hamiltonian, Euler-Lagrange equations), we introduce the first attempt to reconcile quantum physics and special relativity with the Klein-Gordon equation.

We demonstrate the limits of this equation in particular the problem it poses with the introduction of conservation of the probability density which leads to negative densities.

The Pauli equation is then introduced and its limitations are explained.
We then introduce the argumentative list of the conditions that a quanto-relativistic equation must respect, which naturally leads to demonstrating the Dirac equation.

From Dirac’s equation we particularly explain that it naturally contains the spin of the electron without the need for additional conditions.

Finally, we completely develop Dirac’s equation by transforming it to the order v2/c2 so as to reveal all the terms used in atomic physics for the study of atoms under electric and magnetic field.

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