Lamb shift

From Wikipedia, the free encyclopedia

Quantum field theory
Feynman diagram
History
Background Field theory Gauge theory Poincaré symmetry Quantum mechanics Spontaneous symmetry breaking
Symmetries Charge conjugation Crossing Parity Time reversal Symmetry in quantum mechanics
Tools Anomaly Effective field theory Expectation value Faddeev–Popov ghosts Feynman diagram Lattice gauge theory LSZ reduction formula Partition function Propagator Quantization Regularization Renormalization Vacuum state Wick's theorem Wightman axioms
Equations Dirac equation Klein–Gordon equation Proca equations Wheeler–DeWitt equation Bargmann–Wigner equations
Standard Model Electroweak interaction Higgs mechanism Quantum chromodynamics Quantum electrodynamics Yang–Mills theory
Incomplete theories Quantum gravity String theory Supersymmetry Technicolor Theory of everything
Scientists C. D. Anderson P. W. Anderson Bethe Bjorken Bogoliubov Brout Callan Coleman DeWitt Dirac Dyson Englert Fermi Feynman Fierz Fock Fröhlich Glashow Gell-Mann Gross Guralnik Heisenberg Higgs Haag Hagen 't Hooft Kendall Kibble Lamb Landau Lee Majorana Mills Nambu Nishijima Parisi Polyakov Salam Schwinger Skyrme Sudarshan Tomonaga Veltman Ward Weinberg Weyl Wilczek Wilson Yang Yukawa
v t e

Fine structure of energy levels in hydrogen - relativistic corrections to the Bohr model

In physics, the Lamb shift, named after Willis Lamb (1913–2008), is a small difference in energy between two energy levels ²S_{1/2 and ²P_{1/2 (in term symbol notation) of the hydrogen atom in quantum electrodynamics (QED). According to the Dirac equation, the ²S_{1/2 and ²P_{1/2 orbitals should have the same energies. However, the interaction between the electron and the vacuum (which is not accounted for by the Dirac equation) causes a tiny energy shift which is different for states ²S_{1/2 and ²P_{1/2. Lamb and Robert Retherford measured this shift in 1947,^[1] and this measurement provided the stimulus for renormalization theory to handle the divergences. It was the harbinger of modern quantum electrodynamics developed by Julian Schwinger, Richard Feynman, Ernst Stueckelberg and Shinichiro Tomonaga. Lamb won the Nobel Prize in Physics in 1955 for his discoveries related to the Lamb shift.}}}}}}

Derivation

This heuristic derivation of the electrodynamic level shift following Welton is from Quantum Optics.^[2]

The fluctuation in the electric and magnetic fields associated with the QED vacuum perturbs the Coulomb potential due to the atomic nucleus. This perturbation causes a fluctuation in the position of the electron, which explains the energy shift. The difference of potential energy is given by

$\Delta V=V({\vec {r}}+\delta {\vec {r}})-V({\vec {r}})=\delta {\vec {r}}\cdot \nabla V+{\frac {1}{2}}(\delta {\vec {r}}\cdot \nabla )^{2}V({\vec {r}})+...$

Since the fluctuations are isotropic,

$\langle \delta {\vec {r}}\rangle _{{vac}}=0$

$\langle (\delta {\vec {r}}\cdot \nabla )^{2}\rangle _{{vac}}={\frac {1}{3}}\langle (\delta {\vec {r}})^{2}\rangle _{{vac}}\nabla ^{2}$ .

So we can obtain

$\langle \Delta V\rangle ={\frac {1}{6}}\langle (\delta {\vec {r}})^{2}\rangle _{{vac}}\left\langle \nabla ^{2}\left({\frac {-e^{2}}{4\pi \epsilon _{0}r}}\right)\right\rangle _{{at}}$ .

The classical equation of motion for the electron displacement (δr)_k→ induced by a single mode of the field of wave vector k→ and frequency ν is

$m{\frac {d^{2}}{dt^{2}}}(\delta r)_{{{\vec {k}}}}=-eE_{{{\vec {k}}}}$ ,

and this is valid only when the frequency ν is greater than ν₀ in the Bohr orbit, ν > πc/a₀.

For the field oscillating at ν,

$\delta r(t)\cong \delta r(0)e^{{-i\nu t}}+c.c.$ ,

therefore

$(\delta r)_{{{\vec {k}}}}\cong {\frac {e}{mc^{2}k^{2}}}E_{{{\vec {k}}}}={\frac {e}{mc^{2}k^{2}}}{\mathcal {E}}_{{{\vec {k}}}}(a_{{{\vec {k}}}}e^{{-i\nu t+i{\vec {k}}\cdot {\vec {r}}}}+h.c.)$ .

By the summation over all ${\vec {k}}$ ,

$\langle (\delta {\vec {r}})^{2}\rangle _{{vac}}=\sum _{{{\vec {k}}}}\left({\frac {e}{mc^{2}k^{2}}}\right)^{2}\langle 0|(E_{{{\vec {k}}}})^{2}|0\rangle =\sum _{{{\vec {k}}}}\left({\frac {e}{mc^{2}k^{2}}}\right)^{2}\left({\frac {\hbar ck}{2\epsilon _{0}\Omega }}\right)$ ,

where $\Omega$ is some large normalization volume (the volume of the hypothetical "box" containing the hydrogen atom) and

${\mathcal {E}}_{{{\vec {k}}}}=(\hbar ck/2\epsilon _{0}\Omega )^{{1/2}}$ .

The summation is changed into an integral because of the continuity of k→, $\sum _{{{\vec {k}}}}\rightarrow 2{\frac {\Omega }{(2\pi )^{3}}}\int d^{3}k$ , so that

$\langle (\delta {\vec {r}})^{2}\rangle _{{vac}}=2{\frac {\Omega }{(2\pi )^{3}}}4\pi \int dkk^{2}\left({\frac {e}{mc^{2}k^{2}}}\right)^{2}\left({\frac {\hbar ck}{2\epsilon _{0}\Omega }}\right)={\frac {1}{2\epsilon _{0}\pi ^{2}}}\left({\frac {e^{2}}{\hbar c}}\right)\left({\frac {\hbar }{mc}}\right)^{2}\int {\frac {dk}{k}}$ .

This result diverges when there is no limit about the integral. But this method is valid only when ν > πc/a₀, or equivalently k > π/a₀. It is also valid only for wavelengths longer than the Compton wavelength, or equivalently k < mc/ħ. Therefore we can choose the upper and lower limit of the integral and these limits make the result converge.

$\langle (\delta {\vec {r}})^{2}\rangle _{{vac}}\cong {\frac {1}{2\epsilon _{0}\pi ^{2}}}\left({\frac {e^{2}}{\hbar c}}\right)\left({\frac {\hbar }{mc}}\right)^{2}\ln {\frac {4\epsilon _{0}\hbar c}{e^{2}}}$ .

For the atomic orbital and the Coulomb potential,

$\left\langle \nabla ^{2}\left({\frac {-e^{2}}{4\pi \epsilon _{0}r}}\right)\right\rangle _{{at}}={\frac {-e^{2}}{4\pi \epsilon _{0}}}\int d{\vec {r}}\psi ^{*}({\vec {r}})\nabla ^{2}\left({\frac {1}{r}}\right)\psi ({\vec {r}})={\frac {e^{2}}{\epsilon _{0}}}|\psi (0)|^{2}$ ,

since we know that

$\nabla ^{2}\left({\frac {1}{r}}\right)=-4\pi \delta ({\vec {r}})$ .

For p orbitals, the nonrelativistic wave function vanishes at the origin, so there is no energy shift. But for s orbitals there is some finite value at the origin,

$\psi _{{2S}}(0)={\frac {1}{(8\pi a_{0}^{3})^{{1/2}}}}$ ,

where the Bohr radius is

$a_{0}={\frac {4\pi \epsilon _{0}\hbar ^{2}}{me^{2}}}$ .

Therefore

$\left\langle \nabla ^{2}\left({\frac {-e^{2}}{4\pi \epsilon _{0}r}}\right)\right\rangle _{{at}}={\frac {e^{2}}{\epsilon _{0}}}|\psi _{{2S}}(0)|^{2}={\frac {e^{2}}{8\pi \epsilon _{0}a_{0}^{3}}}$ .

Finally, the difference of the potential energy becomes

$\langle \Delta V\rangle ={\frac {4}{3}}{\frac {e^{2}}{4\pi \epsilon _{0}}}{\frac {e^{2}}{4\pi \epsilon _{0}\hbar c}}\left({\frac {\hbar }{mc}}\right)^{2}{\frac {1}{8\pi a_{0}^{3}}}\ln {\frac {4\epsilon _{0}\hbar c}{e^{2}}}$ .

This shift is about 1 GHz, very similar with the observed energy shift.

Experimental work

In 1947 Willis Lamb and Robert Retherford carried out an experiment using microwave techniques to stimulate radio-frequency transitions between ²S_{1/2 and ²P_{1/2 levels of hydrogen.^[3] By using lower frequencies than for optical transitions the Doppler broadening could be neglected (Doppler broadening is proportional to the frequency). The energy difference Lamb and Retherford found was a rise of about 1000 MHz of the ²S_{1/2 level above the ²P_{1/2 level.}}}}

This particular difference is a one-loop effect of quantum electrodynamics, and can be interpreted as the influence of virtual photons that have been emitted and re-absorbed by the atom. In quantum electrodynamics the electromagnetic field is quantized and, like the harmonic oscillator in quantum mechanics, its lowest state is not zero. Thus, there exist small zero-point oscillations that cause the electron to execute rapid oscillatory motions. The electron is "smeared out" and the radius is changed from r to r + δr.

The Coulomb potential is therefore perturbed by a small amount and the degeneracy of the two energy levels is removed. The new potential can be approximated (using atomic units) as follows:

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

The Lamb shift itself is given by

$\Delta E_{{\mathrm {Lamb}}}=\alpha ^{5}m_{e}c^{2}{\frac {k(n,0)}{4n^{3}}}\ {\mathrm {for}}\ \ell =0\,$

with k(n, 0) around 13 varying slightly with n, and

$\Delta E_{{\mathrm {Lamb}}}=\alpha ^{5}m_{e}c^{2}{\frac {1}{4n^{3}}}\left[k(n,\ell )\pm {\frac {1}{\pi (j+{\frac {1}{2}})(\ell +{\frac {1}{2}})}}\right]\ {\mathrm {for}}\ \ell \neq 0\ {\mathrm {and}}\ j=\ell \pm {\frac {1}{2}},$

with k(n,ℓ) a small number (< 0.05).

For a derivation of ΔE_Lamb see for example:^[4]

Lamb shift in the hydrogen spectrum

In 1947, Hans Bethe was the first to explain the Lamb shift in the hydrogen spectrum, and he thus laid the foundation for the modern development of quantum electrodynamics. The Lamb shift currently provides a measurement of the fine-structure constant α to better than one part in a million, allowing a precision test of quantum electrodynamics.

A different perspective relates Zitterbewegung to the Lamb shift.^[5]

References

↑ G Aruldhas (2009). "§15.15 Lamb Shift". Quantum Mechanics (2nd ed.). Prentice-Hall of India Pvt. Ltd. p. 404. ISBN 81-203-3635-6.
↑ Marlan Orvil Scully & Muhammad Suhail Zubairy (1997). Quantum optics. Cambridge UK: Cambridge University Press. pp. 13–16. ISBN 0-521-43595-1.
↑ Lamb, Willis E.; Retherford, Robert C. (1947). "Fine Structure of the Hydrogen Atom by a Microwave Method". Physical Review 72 (3): 241–243. Bibcode:1947PhRv...72..241L. doi:10.1103/PhysRev.72.241. Cite uses deprecated parameters (help)
↑ Bethe, H.A.& Salpeter, E.E. (1957). Quantum Mechanics of One- and Two-Electron Atoms. Springer. p. 103.
↑ Henning Genz (2002). Nothingness: the science of empty space. Reading MA: Oxford: Perseus. p. 245 ff. ISBN 0-7382-0610-5.

External links

v t e Quantum electrodynamics

anomalous magnetic dipole moment Bhabha scattering bremsstrahlung Compton scattering electron Gupta–Bleuler formalism Møller scattering photon positron positronium QED vacuum self-energy vacuum polarization virtual particles vertex function Ward–Takahashi identity ξ gauge

v t e Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum Yang–Mills theory Schwinger model Sine–Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model

Particles in physics

Elementary

Fermions

Quarks	u u d d c c s s t t b b

Leptons	e− e+ μ− μ+ τ− τ+ ν e ν e ν μ ν μ ν τ ν τ

Bosons

Gauge	γ g W± · Z

Scalar	H0

Others

Ghosts

Hypothetical

Superpartners

Gauginos	Gluino Gravitino

Others	Axino Chargino Higgsino Neutralino Sfermion

Others

Composite

Hadrons

Baryons / Hyperons	N p n Δ Λ Σ Ξ Ω

Mesons / Quarkonia	π ρ η η′ φ ω J/ψ ϒ θ K B D T

Others

Atomic nuclei
Atoms
Diquarks
Exotic atoms
- Positronium
- Muonium
- Tauonium
- Onia
Superatoms
Molecules

Hypothetical

Exotic
hadrons

Exotic baryons	Dibaryon Pentaquark Skyrmion

Exotic mesons	Glueball Tetraquark

Others

Quasiparticles

Lists

Wikipedia books

Hadronic Matter
Particles of the Standard Model
Leptons
Quarks

Physics portal

This article is issued from Wikipedia. The text is available under the Creative Commons Attribution/Share Alike; additional terms may apply for the media files.

Lamb shift

Derivation

Experimental work

Lamb shift in the hydrogen spectrum

See also

References

Further reading

External links