Fine structure

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In atomic physics, the fine structure describes the splitting of the spectral lines of atoms.

The gross structure of line spectra is the number of lines and their placement. This is determined by the differences in the energy levels of the various atomic orbitals. However, on closer examination, each line exhibits a detailed fine structure. This structure is due to small interactions that give small shifts and splittings of the energy levels. They may be analyzed by means of perturbation theory. The fine structure of hydrogen is actually two separate corrections to the Bohr energies: one due to the relativistic motion of the electron, and the other due to spin-orbit coupling.

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Classically, the kinetic energy term of the Hamiltonian is:

T=\frac{p^{2}}{2m}

However, when considering special relativity, we must use a relativistic form of the kinetic energy,

T=\sqrt{p^{2}c^{2}+m^{2}c^{4}}-mc^{2}

where the first term is the total relativistic energy, and the second term is the rest energy of the electron. Expanding this we find

T=\frac{p^{2}}{2m}-\frac{p^{4}}{8m^{3}c^{2}}+\dots

Then, the first order correction to the Hamiltonian is

H'=-\frac{p^{4}}{8m^{3}c^{2}}

Using this as a perturbation, we can calculate the first order energy corrections due to relativistic effects.

E_{n}^{(1)}=\langle\psi^{0}\vert H'\vert\psi^{0}\rangle=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert p^{4}\vert\psi^{0}\rangle=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert p^{2}p^{2}\vert\psi^{0}\rangle

where ψ0 is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see

H^{0}\vert\psi^{0}\rangle=E_{n}\vert\psi^{0}\rangle

\left(\frac{p^{2}}{2m}-V\right)\vert\psi^{0}\rangle=E_{n}\vert\psi^{0}\rangle

p^{2}\vert\psi^{0}\rangle=2m(E_{n}-V)\vert\psi^{0}\rangle

We can use this result to further calculate the relativistic correction:

E_{n}^{(1)}=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert p^{2}p^{2}\vert\psi^{0}\rangle

E_{n}^{(1)}=-\frac{1}{8m^{3}c^{2}}\langle\psi^{0}\vert (2m)^{2}(E_{n}-V)^{2}\vert\psi^{0}\rangle

E_{n}^{(1)}=-\frac{1}{2mc^{2}}(E_{n}^{2}-2E_{n}\langle V\rangle +\langle V^{2}\rangle )

For the hydrogen atom, V=\frac{e^{2}}{r}, \langle V\rangle=\frac{e^{2}}{a_{0}n^{2}}, and \langle V^{2}\rangle=\frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}} where a0 is the Bohr Radius, n is the principal quantum number and l is the azimuthal quantum number. Therefore the relativistic correction for the hydrogen atom is

E_{n}^{(1)}=-\frac{1}{2mc^{2}}\left(E_{n}^{2}-2E_{n}\frac{e^{2}}{a_{0}n^{2}} +\frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}}\right)=-\frac{E_{n}^{2}}{2mc^{2}}\left(\frac{4n}{l+1/2}-3\right)

The Spin-Orbit correction arises when we shift from the standard frame of reference (where the electron orbits the nucleus) into one where the electron is stationary and the nucleus instead orbits it. In this case the orbiting nucleus functions as an effective current loop, which in turn will generate a magnetic field. However, the electron itself has a magnetic moment due to its intrinsic angular momentum. The two magnetic vectors, \vec B and \vec\mu_s couple together so that there is a certain energy cost depending on their relative orientation. This gives rise to the energy correction of the form

\Delta E_{SO} = \xi (r)\vec L \cdot \vec S

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