Uehling potential

The Uehling potential , according to Edwin Albrecht Uehling , is the modification of the Coulomb potential of electrostatics through the effects of quantum electrodynamics . The effects of this modification are also called the Uehling effect . The corrections by Uehling are meaningless in everyday practice, but they provide a measurable part in the Lamb shift of the energies for the electrons in the potential of an atomic nucleus and thus for the position and splitting of the spectral lines .

Feynman diagram of a virtual particle-antiparticle loop (lines with arrows) as self-energy correction of a photon (wavy line)

The corrections by Uehling take into account that the electric field of a point charge does not exert a long-range effect , but rather an interaction takes place via exchange particles, the photons . In quantum field theory, due to the energy-time uncertainty, a single photon can briefly form a virtual particle-antiparticle pair, so that the potential of the point charge is influenced. This effect is called vacuum polarization , as it makes the vacuum appear like a polarizable medium. By far the dominant contribution comes from the lightest charged elementary particle, the electron.

The Uehling potential is:

${\displaystyle V(r)=-Z\alpha \hbar c{\frac {1}{r}}\left(1+{\frac {2\alpha }{3\pi }}\int _{1}^{\infty }\mathrm {d} x\,e^{-4\pi {\frac {rx}{\lambda }}}{\frac {2x^{2}+1}{2x^{4}}}{\sqrt {x^{2}-1}}\right)+{\mathcal {O}}(\alpha ^{3})}$

Here designated

• ${\displaystyle Z}$the number of charges in units of the elementary charge ${\displaystyle e}$
• ${\displaystyle \alpha }$the fine structure constant
• ${\displaystyle \hbar }$the reduced Planck quantum of action
• ${\displaystyle c}$the speed of light
• ${\displaystyle \lambda }$the Compton wavelength of the electron
• ${\displaystyle r}$ the distance between the point charge and the place under consideration

The first term is the Coulomb potential of classical electrostatics, in terms of classical physics with the electric field constant ${\displaystyle \varepsilon _{0}}$

${\displaystyle V_{\text{C}}(r)=-Z\alpha \hbar c{\frac {1}{r}}=-Z{\frac {e^{2}}{4\pi \varepsilon _{0}}}{\frac {1}{r}}}$

and the integral term the correction by quantum electrodynamics. This is of the second order in the fine structure constant, since in quantum field theoretic perturbation theory it is induced by a loop (see figure). The Uehling term is positive for all distances, so it leads to an amplification of the potential.

The integral cannot be represented by elementary functions, but the exponential function in this obviously leads to a strong suppression of the effect for large distances, the relevant length scale being the Compton wavelength of the electron. To illustrate the order of magnitude, it is only a fraction of an atomic diameter; at a distance from Bohr's radius , the most likely location of the electron around a hydrogen atom, the deviation is only in the order of magnitude of 10 ⁻¹²⁵ . For long distances the potential can also be used as

${\displaystyle V(r)\approx -Z\alpha \hbar c{\frac {1}{r}}\left(1+{\frac {\alpha }{8\pi ^{2}{\sqrt {2}}}}\left({\frac {\lambda }{r}}\right)^{3/2}e^{-4\pi {\frac {r}{\lambda }}}\right)+{\mathcal {O}}(\alpha ^{3})}$

to be approached.

On the other hand, the integral does not converge in the limit of small distances , so that the Uehling potential there generates a measurable deviation from the Coulomb potential. The following applies to small distances ${\displaystyle r\to 0}$

${\displaystyle V(r)\approx -Z\alpha \hbar c{\frac {1}{r}}\left(1+{\frac {2\alpha }{3\pi }}\left(\ln {\frac {\lambda }{2\pi r}}-\gamma -{\frac {5}{6}}\right)\right)+{\mathcal {O}}(\alpha ^{3})}$

with the Euler-Mascheroni constant . ${\displaystyle \gamma }$

Influence on energy levels in the atom

Since the Uehling potential only makes a significant contribution for very small distances around the nucleus, it mainly influences the energy of the s orbitals . Quantum mechanical perturbation theory can be used to calculate this influence . In contrast to the potential itself, the results can be presented analytically in a closed manner, since the integrations that occur over the distance and the integration parameters interchange. The energy corrections for the energy levels degenerate according to a quantum mechanical calculation and are in leading order with the electron mass : ${\displaystyle r}$ ${\displaystyle x}$ ${\displaystyle 2S_{1/2}}$ ${\displaystyle 2P_{1/2}}$ ${\displaystyle m}$

${\displaystyle {\begin{aligned}\Delta E(2S_{1/2})&=-{\frac {\alpha ^{5}}{30\pi }}mc^{2}\approx -1{,}1\cdot 10^{-7}\,{\text{eV}}\\\Delta E(2P_{1/2})&=-{\frac {3\alpha ^{7}}{2240\pi }}mc^{2}\approx -2{,}4\cdot 10^{-13}\,{\text{eV}}\end{aligned}}}$

Since the wave function of the s orbitals does not disappear at the origin, the Uehling potential contributes to the order , i.e. to the Lamb shift, while it becomes less important for orbitals with a higher angular momentum quantum number . ${\displaystyle \alpha ^{5}}$

For muonic hydrogen , the Uehling effect is central: In contrast to other variables such as the splitting by the fine structure , which scale together with the mass of the muon, which is around 200 times heavier, the light electron mass continues to be the decisive size scale for the Uehling potential; the energy corrections are then of the same order of magnitude , i.e. around 40,000 times greater than in standard hydrogen. ${\displaystyle m_{\mu }^{3}/m_{e}^{2}c^{2}\alpha ^{5}}$

literature

• Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 (English).
• VB Berestetskii, EM Lifshitz and LP Pitaevskii: Quantum Electrodynamics . 2nd Edition. Pergamon Press, Oxford New York Toronto Sydney Paris Frankfurt 1982 (English, Russian: Kvantovaya elektrodinamika . Translated by JB Sykes and JS Bell).
• Edwin A. Uehling: Polarization effects in the positron theory . In: Phys. Rev. Band 48 , no. 1 , 1935, p. 55-63 (English).