NationMaster - Encyclopedia: Fine structure

In atomic physics, the fine structure describes the splitting of the spectral lines of atoms. Atomic physics (or atom physics) is the field of physics that studies atoms as isolated systems comprised of electrons and an atomic nucleus. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... Properties For alternative meanings see atom (disambiguation). ...

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. A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... An atomic orbital is the description of the behavior of an electron in an atom according to quantum mechanics. ... In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ... The Bohr model of the hydrogen atom, where negatively charged electrons confined to atomic shells encircle a small positively charged atomic nucleus, and that an electron jump between orbits must be accompanied by an emitted or absorbed amount of electromagnetic energy hÎ½. The orbits that the electrons travel in are... In quantum mechanics, the orbital and spin angular momentum of bodies can interact in angular momentum coupling. ...

1 Relativistic corrections
2 Spin-orbit coupling
3 References
4 External links

Relativistic corrections

Classically, the kinetic energy term of the Hamiltonian is: The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). ...

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

However, when considering special relativity, we must use a relativistic form of the kinetic energy, The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest...

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

where the first term is the total relativistic energy, and the second term is the rest energy of the electron. Expanding this we find The rest energy of a particle is its energy when it is not moving relative to a given inertial reference frame. ...

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

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.

$E_{n}^{(1)}=langlepsi^{0}vert H'vertpsi^{0}rangle=-frac{1}{8m^{3}c^{2}}langlepsi^{0}vert p^{4}vertpsi^{0}rangle=-frac{1}{8m^{3}c^{2}}langlepsi^{0}vert p^{2}p^{2}vertpsi^{0}rangle$

where $ψ 0$ is the unperturbed wave function. Recalling the unperturbed Hamiltonian, we see

$H^{0}vertpsi^{0}rangle=E_{n}vertpsi^{0}rangle$

$left(frac{p^{2}}{2m}-Vright)vertpsi^{0}rangle=E_{n}vertpsi^{0}rangle$

$p^{2}vertpsi^{0}rangle=2m(E_{n}-V)vertpsi^{0}rangle$

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

$E_{n}^{(1)}=-frac{1}{8m^{3}c^{2}}langlepsi^{0}vert p^{2}p^{2}vertpsi^{0}rangle$

$E_{n}^{(1)}=-frac{1}{8m^{3}c^{2}}langlepsi^{0}vert (2m)^{2}(E_{n}-V)^{2}vertpsi^{0}rangle$

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

For the hydrogen atom, $V=frac{e^{2}}{r}$ , $langle Vrangle=frac{e^{2}}{a_{0}n^{2}}$ , and $langle V^{2}rangle=frac{e^{4}}{(l+1/2)n^{3}a_{0}^{2}}$ where $a 0$ 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 In the Bohr model of the structure of an atom, put forward by Niels Bohr in 1913, electrons orbit a central nucleus. ... In atomic physics, the principal quantum number symbolized as n is the first quantum number of an atomic orbital. ... The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l (lower-case L) is a quantum number for an atomic orbital which determines its orbital angular momentum. ...

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

Spin-orbit coupling

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 $vecmu_s$ 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 In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... This article or section is in need of attention from an expert on the subject. ... e- redirects here. ... Look up nucleus in Wiktionary, the free dictionary. ... In physics, spin is an intrinsic angular momentum associated with microscopic particles. ...

$Delta E_{SO} = &# 0;(r)vec L cdot vec S$

References

Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0-13-805326-X.
Liboff, Richard L. (2002). Introductory Quantum Mechanics. Addison-Wesley. ISBN 0-8053-8714-5.

Richard L. Liboff is a U.S. physicist who has authored five books and nearly 150 other publications in variety of fields, including plasma physics, planetary physics, cosmology, quantum chaos, and quantum billiards. ...

External links

Hyperphysics: Fine Structure

Categories: Atomic physics | Atomic, molecular, and optical physics stubs

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Relativistic corrections var ACE_AR = {Site: '756743', Size: '300250'};

Spin-orbit coupling

References

External links

Relativistic corrections