Fine structure

The interference pattern obtained in the Fabry-Perot interferometer from a light source in the form of highly chilled deuterium and showing fine line splitting.

Fine structure ( multiplet splitting ) is a phenomenon in atomic physics that describes the splitting of spectral lines (energy levels, spectral terms ) of an atom .

The macroscopic structure of spectral lines is the number of lines and their location. It is determined by the difference in the energy levels of various atomic orbitals . However, with a more detailed study, each line shows its detailed fine structure. This structure is explained by small interactions that slightly shift and split energy levels. They can be analyzed by perturbation theory methods. The fine structure of the hydrogen atom actually represents two independent corrections to the Bohr energies : one due to the relativistic motion of the electron, and the second due to the spin-orbit interaction .

Content

Relativistic Amendments

In classical theory, the kinetic term of the Hamiltonian : $T={\frac {p^{2}}{2m}}$ ${\ displaystyle T = {\ frac {p ^ {2}} {2m}}}$

However, given SRT , we must use the relativistic expression for kinetic energy, $T={\sqrt {p^{2}c^{2}+m^{2}c^{4}}}-mc^{2}$ ${\ displaystyle T = {\ sqrt {p ^ {2} c ^ {2} + m ^ {2} c ^ {4}}} - mc ^ {2}}$

where the first term is the total relativistic energy, and the second term is the rest energy of the electron. Putting it in a row, we get

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

Hence, the first-order correction to the Hamiltonian is $H'=-{\frac {p^{4}}{8m^{3}c^{2}}}$ ${\ displaystyle H '= - {\ frac {p ^ {4}} {8m ^ {3} c ^ {2}}}}$

Using this as a perturbation, we can calculate first-order relativistic energy corrections.

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

Where $\psi ^{0}$ ${\ displaystyle \ psi ^ {0}}$ - unperturbed wave function . Remembering the undisturbed Hamiltonian, we see

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

$\left({\frac {p^{2}}{2m}}+U\right)\vert \psi ^{0}\rangle =E_{n}\vert \psi ^{0}\rangle$ ${\ displaystyle \ left ({\ frac {p ^ {2}} {2m}} + U \ right) \ vert \ psi ^ {0} \ rangle = E_ {n} \ vert \ psi ^ {0} \ rangle }$

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

Further, we can use this result to calculate the relativistic correction:

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

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

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

For a hydrogen atom, $U={\frac {e^{2}}{r}}$ ${\ displaystyle U = {\ frac {e ^ {2}} {r}}}$ , $\langle U\rangle ={\frac {e^{2}}{a_{0}n^{2}}}$ ${\ displaystyle \ langle U \ rangle = {\ frac {e ^ {2}} {a_ {0} n ^ {2}}}}$ and $\langle U^{2}\rangle ={\frac {e^{4}}{(l+1/2)n^{3}a_{0}^{2}}}$ ${\ displaystyle \ langle U ^ {2} \ rangle = {\ frac {e ^ {4}} {(l + 1/2) n ^ {3} a_ {0} ^ {2}}}}$ Where $a_{0}$ ${\ displaystyle a_ {0}}$ - Borovsky radius , $n$ ${\ displaystyle n}$ Is the main quantum number and $l$ ${\ displaystyle l}$ Is the orbital quantum number . Therefore, the relativistic correction for the hydrogen atom is

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

Spin Orbit Relationship

The spin-orbit correction appears when we move from the standard reference system (where the electron flies around the nucleus) to the system where the electron is at rest and the nucleus flies around it. In this case, the moving core is an effective current loop , which in turn creates a magnetic field . However, the electron itself has a magnetic moment due to the spin. Two magnetic vectors, ${\vec {B}}$ ${\ displaystyle {\ vec {B}}}$ and ${\vec {\mu }}_{s}$ ${\ displaystyle {\ vec {\ mu}} _ {s}}$ are linked together so that a certain energy appears, depending on their relative orientation. So there is an energy correction of the form $\Delta E_{SO}=\xi (r){\vec {L}}\cdot {\vec {S}}$ ${\ displaystyle \ Delta E_ {SO} = \ xi (r) {\ vec {L}} \ cdot {\ vec {S}}}$

Spontaneous production of electron-positron pairs

Spontaneous production of electron-positron pairs near an electron leads to the fact that the localization of an electron in an atom in a region shorter than its Compton wavelength $\Delta x={\frac {\hbar }{mc}}$ ${\ displaystyle \ Delta x = {\ frac {\ hbar} {mc}}}$ impossible and as a result there is a quadratic fluctuation of the electron position $\Delta x^{2}$ ${\ displaystyle \ Delta x ^ {2}}$ . As a result, the potential energy of the electron changes inside the nucleus. The energy shift is: $\Delta E_{ep}=m(Z\alpha )^{4}$ ${\ displaystyle \ Delta E_ {ep} = m (Z \ alpha) ^ {4}}$ where $m$ ${\ displaystyle m}$ is the mass of the electron, $Z$ ${\ displaystyle Z}$ - effective charge of the nucleus, $\alpha$ ${\ displaystyle \ alpha}$ - constant fine structure. ^[one]

Literature

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

Links

Hyperphysics: Fine Structure

Notes

↑ V. Tirring. Principles of quantum electrodynamics. M., Higher School, 1964. - p. 18-19

[1] V. Tirring. Principles of quantum electrodynamics. M., Higher School, 1964. - p. 18-19