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.

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 $\psi^{0}$ 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 $a_{0}$ 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

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

Spin-Orbit Coupling

Classically, orbiting charges possess a magnetic dipole moment, and this holds in quantum mechanics also. Slightly more surprisingly, the intrinsic angular momentum of particles due to spin gives them a magnetic moment; thus the 'spin-half' electron acts like a little magnet. Since it is orbiting the positively charged nucleus, it 'sees' a current, and hence a magnetic field.

References

External Links

Hyperphysics: Fine Structure

Atomic physics

Feinstruktur (Physik)

Gourt Home::Gourt :: Fine structure

Relativistic Corrections

Spin-Orbit Coupling

References

External Links