There is one last term in the non-relativistic expansion of the Dirac equation. When weak external magnetic fields are applied, the spin–orbit coupling contributes to the Zeeman effect. The gross structure assumes the kinetic energy term of the Hamiltonian takes the same form as in classical mechanics, which for a single electron means This section discusses the analytical solutions for the hydrogen atom as the problem is analytically solvable and is the base model for energy level calculations in more complex atoms. These corrections can also be obtained from the non-relativistic limit of the Dirac equation, since Dirac's theory naturally incorporates relativity and spin interactions. To perform this calculation one must add the three corrective terms to the Hamiltonian: the leading order relativistic correction to the kinetic energy, the correction due to the spin–orbit coupling, and the Darwin term coming from the quantum fluctuating motion or zitterbewegung of the electron. The fine structure energy corrections can be obtained by using perturbation theory. The scale of the fine structure splitting relative to the gross structure energies is on the order of ( Zα) 2, where Z is the atomic number and α is the fine-structure constant, a dimensionless number equal to approximately 1/137. However, a more accurate model takes into account relativistic and spin effects, which break the degeneracy of the energy levels and split the spectral lines. For a hydrogenic atom, the gross structure energy levels only depend on the principal quantum number n. The gross structure of line spectra is the line spectra predicted by the quantum mechanics of non-relativistic electrons with no spin. Morley in 1887, laying the basis for the theoretical treatment by Arnold Sommerfeld, introducing the fine-structure constant. It was first measured precisely for the hydrogen atom by Albert A. We can calculate the energy of any level (En) in hydrogen atom from the relation: En 13.6 ( e V ) / n², Where: Energy (Joule) Energy (e V) × Charge of the electron (Coulomb) 1 eV 1.6 × 1019 J. In atomic physics, the fine structure describes the splitting of the spectral lines of atoms due to electron spin and relativistic corrections to the non-relativistic Schrödinger equation.
Interference fringes, showing fine structure (splitting) of a cooled deuterium source, viewed through a Fabry–Pérot interferometer.
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