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FINE STRUCTURE

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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Relativistic Corrections

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)

Spin-Orbit Coupling

Classically, orbiting charges possess a magnetic dipole moment, and this holds in quantum mechanics also. Slightly more surprising, the intrinsic angular momentum of particles due to spin gives them a magnetic moment. Fermions, such as electrons and protons, compose the "stuff" of matter and have half-integer spin (the unit being \hbar\equiv\frac{h}{2\pi} where h is Planck's Constant). Bosons, the particles that carry the forces that hold matter together, such as photons and gluons, have non-zero integer spin. The vector of the magnetic moment in an electron, derived from the flux of charge, occupies only one dimension in respect to all electrons. A positive scalar quantitity of this vector is called "spin-up", while the negative is "spin-down". Only the theoretical Higgs Boson has no spin at all.

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References

  • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.
  • Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0805387145.

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