Lamb shift

Lamb shift as one of several splits of the energy levels of the hydrogen atom

The Lamb shift (also Lamb shift ) is an effect in quantum physics that was discovered in 1947 by Willis Eugene Lamb and Robert Curtis Retherford (1912–1981).

The experiment showed that two atomic states in the hydrogen atom, which according to the Dirac theory of relativistic quantum mechanics should have exactly the same energies, have a - very small - energy difference. This discovery laid the foundation for quantum electrodynamics . Lamb was awarded the Nobel Prize in Physics for this in 1955 . The Nobel Prize is related to the effect on the hydrogen atom, but the Lamb shift is a general quantum electrodynamic effect.

description

The Dirac equation states that states with the same principal quantum number and the same total angular momentum quantum number (orbital angular momentum plus spin) in hydrogen or in hydrogen-like atoms are degenerate with regard to the secondary quantum number , i.e. H. have the same energy. The lowest states, which should therefore be degenerate, are the states and , both of which have the quantum numbers and . ${\displaystyle n}$ ${\displaystyle j}$ ${\displaystyle \ell }$ ${\displaystyle 2s_{1/2}}$ ${\displaystyle (\ell =0)}$ ${\displaystyle 2p_{1/2}}$ ${\displaystyle (\ell =1)}$ ${\displaystyle n=2}$ ${\displaystyle j={\tfrac {1}{2}}}$

Lamb and Retherford generated a beam of hydrogen atoms in the 2s _1/2 state and exposed it to microwave radiation of 2395 MHz. This raised the atoms to the 2p _3/2 state and from there fell to the 2p _1/2 state. An external magnetic field caused the energy levels to split up through the Zeeman effect . By varying the magnetic field, they were able to determine the energies of the transitions very precisely and found that the state 2p _1/2 is 4.37  μeV lower than 2s _1/2 , corresponding to a frequency difference Δν = 1058 MHz. Compared to the energy of the two levels of −3.4 eV, this is a very small correction (by a factor of 10 even smaller than the fine structure split between 2p _1/2 and 2p _3/2 ), but it is of fundamental importance.

Explanation

Contributions to the Lamb shift in the H atom.

Contribution	2p 1/2	2s 1/2
Self-energy of the electron	4.07 MHz	1015.52 MHz
Vacuum polarization	0 MHz	−27.13 MHz
abnormal magnetic moment	−16.95 MHz	50.86 MHz

The first calculations for the Lamb shift were carried out by Hans Bethe , followed by Richard Feynman and Julian Schwinger . Three quantum electrodynamic effects make the largest contribution: the self-energy of the electron, vacuum polarization and the anomalous magnetic moment.

Self-energy

The self-energy of the electron has the largest share in the Lamb shift , i.e. H. its interaction with vacuum fluctuations . In accordance with Heisenberg's uncertainty principle , virtual photons are absorbed and emitted from the vacuum field. The resulting movement (cf. also trembling movement ) changes the potential acting on the electron over time. The effect becomes relevant close to the center of the atom , especially within the nucleus, where the potential deviates from the Coulomb shape. This mainly concerns electrons with angular momentum quantum numbers (s-states), whose probability of being in the nucleus is small but relevant, while for the wave function of the electron in the center it is zero. s electrons are thus bound a little weaker. ${\displaystyle (r=0)}$ ${\displaystyle \ell =0}$ ${\displaystyle \ell >0}$

Therefore a small correction is added to the calculation of the potential energy , which can be written approximately as follows: ${\displaystyle \delta r}$

${\displaystyle \langle E_{\mathrm {pot} }\rangle =-{\frac {Ze^{2}}{4\pi \varepsilon _{0}}}\left\langle {\frac {1}{r+\delta r}}\right\rangle }$

with atomic number , elementary charge , electric field constant and distance . ${\displaystyle Z}$ ${\displaystyle e}$ ${\displaystyle \varepsilon _{0}}$ ${\displaystyle r}$

Vacuum polarization

Another contribution to the Lamb shift can be traced back to the vacuum polarization . By creating and destroying virtual particle pairs, the vacuum behaves like a dielectric medium that shields the charge on the core. Its effective charge is increased very close to the core, which means that the electrical potential is lower ( Uehling effect ). Again, mainly s electrons are affected by this.

The vacuum polarization - with the opposite sign - contributes hardly more than 2% to the total effect (in muonic atoms the proportion is larger), but the theoretical calculations and experiments were so precise that this contribution and thus the vacuum polarization could be confirmed.

Anomalous magnetic moment

Another contribution results from the anomalous magnetic moment of the electron.

Overall effect

There are further amounts of higher order (i.e. described by higher powers of the fine structure constant α).

The Lamb shift results as follows:

${\displaystyle \Delta E_{\mathrm {Lamb} }(n,\ell =0,j={\tfrac {1}{2}})=(Z\alpha )^{4}{\frac {4\alpha }{3\pi n^{3}}}m_{e}c^{2}\left(\ln \left({\frac {m_{e}c^{2}}{2R_{y}}}\right)-\gamma (n,\ell ,j)+{\frac {19}{30}}\right)}$

respectively:

${\displaystyle \Delta E_{\mathrm {Lamb} }(n,\ell \neq 0,j)=(Z\alpha )^{4}{\frac {4\alpha }{3\pi n^{3}}}m_{e}c^{2}\left(\ln \left((Z\alpha )^{2}{\frac {m_{e}c^{2}}{2R_{y}}}\right)-\gamma (n,\ell ,j)+{\frac {3}{8}}{\frac {j(j+1)-\ell (\ell +1)-{\frac {3}{4}}}{\ell (\ell +1)(2\ell +1)}}\right)}$

There are:

• ${\displaystyle \alpha }$the fine structure constant
• ${\displaystyle m_{e}}$ the mass of the electron
• ${\displaystyle c}$the speed of light
• ${\displaystyle n,\ell ,j}$the principal , minor and total angular momentum quantum number
• ${\displaystyle R_{y}}$the Rydberg energy
• ${\displaystyle \gamma }$ the Bethe logarithm

The Bethe logarithm can be calculated numerically and is for the lowest orbitals

${\displaystyle {\begin{aligned}\gamma (1,0,{\tfrac {1}{2}})&=2{,}98\dots \\\gamma (2,0,{\tfrac {1}{2}})&=2{,}81\dots \\\gamma (2,1,j)&=0{,}03\dots \end{aligned}}}$

With these values, the energy difference between the and orbitals , corresponding to a frequency difference between the spectral lines , is in precise agreement with the experiment. ${\displaystyle \delta E}$ ${\displaystyle 2s_{1/2}}$ ${\displaystyle 2p_{1/2}}$ ${\displaystyle \delta E=4{,}37\cdot 10^{-6}\,{\text{eV}}}$ ${\displaystyle \delta f}$ ${\displaystyle \delta f=1058\,{\text{MHz}}}$

For muonic atoms the effect is much stronger because the orbital radius of the muon is much smaller and the anomalous magnetic moment is greater. Likewise, the effect increases with the charge and radius of the nucleus. In the case of atoms with more than one electron, however, it is superimposed by other effects (shielding of the nuclear charge by the other electrons).

Individual evidence

1. ↑ Retherford was a PhD student ( graduate student ) at Columbia University. But he was already an experienced experimental physicist and previously employed in industry (Westinghouse), where he had dealt with vacuum tubes. In the mid-1950s he became a professor at the University of Wisconsin.
2. ^ Willis E. Lamb, Robert C. Retherford: Fine Structure of the Hydrogen Atom by a Microwave Method . In: Physical Review . tape 72 , no. 3 , 1947, pp. 241-243 , doi : 10.1103 / PhysRev.72.241 .
3. ↑ https://www.nobelprize.org/nobel_prizes/physics/laureates/1955/
4. ↑ ^{a b} Kurt Gottfried, Victor F. Weisskopf: Concepts of Particle Physics, Vol II . Clarendon Press, Oxford 1986, ISBN 978-0-19-503393-9 , pp. 266-270 .
5. ^ Hermann hook, Hans Christoph Wolf: atomic and quantum physics . Springer-Verlag, Berlin Heidelberg 2004, May 15, 2.
6. ^ Robert W. Huff: Simplified Calculation of Lamb Shift Using Algebraic Techniques . In: Phys. Rev. Band 186 , no. 5 , 1969, p. 1367-1379 (English).

literature

• Steven Weinberg: The Quantum Theory of Fields Volume I: Foundations . Cambridge University Press, New York 1995 (English).
• Ingolf V. Hertel, Claus-Peter Schulz: Atoms, Molecules and Optical Physics 1 . Springer-Verlag, Berlin Heidelberg 2008.
• Hermann hook, Hans Christoph Wolf: atomic and quantum physics . Springer-Verlag, Berlin Heidelberg 2004.