Fine Structure and Lamb Shift

Introduction

▼ Category:8.421

► Category:Atoms in electric fields

  Atoms

  Atoms in electric fields

  Atoms in Magnetic Fields

  Coherence

  Effects of the Nucleus on Atomic Structure

  Fine Structure and Lamb Shift

  Higher-Order Radiation Processes

  Interaction of an atom with an electromagnetic field

  Old Coherence

  Resonances

  Two-Photon Excitation

Immediately adjacent to Michelson and Morley's announcement of their failure to find the ether in an 1887 issue of the Philosophical Journal is a paper by the same authors reporting that the HFailed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle _{\alpha}} line of hydrogen is actually a doublet, with a separation of 0.33 cm$^{-1}$. In 1915 Bohr suggested that this "fine structure" of hydrogen is a relativistic effect arising from the variation of mass with velocity. Sommerfeld, in 1916, solved the relativistic Kepler problem and using the old quantum theory, as it was later christened, accounted precisely for the splitting.

Sommerfeld's theory gave the lie to Einstein's dictum "The Good Lord is subtle but not malicious", for it gave the right results for the wrong reason: his theory made no provision for electron spin, an essential feature of fine structure. Today, all that is left from Sommerfeld's theory is the fine structure constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha = e^2 /\hbar c} . The theory for the fine structure in hydrogen was provided by Dirac whose relativistic electron theory (1926) was applied to hydrogen by Darwin and Gordon in 1928. They found the following expression for the energy of an electron bound to a proton of infinite mass:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{E}{mc^2} ={\left[ \frac{1}{\sqrt{\frac{\alpha Z}{n-k+\sqrt{k^2-\alpha^2Z^2}}}}\right]^2} }

where $n$ is the principal quantum number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k = j + 1/2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j = \ell \pm 1/2 } . The Dirac equation is not nearly as illuminating as the Pauli equation, which is the approximation to the Dirac equation to the lowest order in v/c.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=mc^2+\frac{p^2}{2m}-\frac{e^2}{r}+H_{FS} }

The first term is the electron's rest energy; the following two terms are the non relativistic Hamiltonian, and the last term, the fine structure interaction, is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{FS}=-\frac{p^4}{8m^3c^2} + \frac{\hbar^2e^2}{2m^2c^2}\frac{1}{r^3}L\cdot S -\frac{\hbar^2}{8m^2c^2}\nabla^2\frac{e^2}{r} }

The relativistic contributions can be described as the kinetic, spin-orbit, and Darwin terms, $H_{\rm {kin}}$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm{so}}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm{Dar}}} , respectively. Each has a straightforward physical interpretation.

Kinetic contribution

Relativistically, the total electron energy is E = Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt {(m c^2 )^2 + (pc )^2 }} . The kinetic energy is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=E-mc^2 = (mc^2 ) {\left[\sqrt{1+\frac{p^2}{m^2c^2}}-1 ~~\right]} =\frac{p^2}{2m} - \frac{1}{8} \frac{p^4}{m^3c^2}+\cdots }

Thus

$H_{\rm {kin}}=-{\frac {1}{8}}{\frac {p^{4}}{m^{3}c^{2}}}$

Spin-Orbit Interaction

According to the Dirac theory the electron has intrinsic angular momentum Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar S} and a magnetic moment Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\mu}_2 = - g_e \mu_B S} . The electron g-factor, $g_{e}=2$ and the Bohr magneton Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu_B=e\hbar /2m c} . As the electron moves through the electric field of the proton it "sees" a motional magnetic field

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{\rm{mot}} = -\frac{v}{c} \times E = - \frac{v}{c} \times \frac{e}{r^3} r = \frac{e\hbar}{mc} \frac{1}{r^3} L }

where $\hbar L=r\times mv$. However, there is another contribution to the effective magnetic field arising from the Thomas precession. The relativistic transformation of a vector between two moving co- ordinate systems which are moving with different velocities involve not only a dilation, but also a rotation (cf Jackson, Classical Electrodynamics). The rate of rotation, the Thomas precession, is

<equation id="EQ_soitwo" />

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \vec{\Omega}_T =\frac{1}{2} \frac{a \times v}{c^2} }

Note that the precession vanishes for co-linear acceleration. However, for a vector fixed in a co-ordinate system moving around a circle, as in the case of the spin vector of the electron as it circles the proton, Thomas precession occurs. From the point of view of an observer fixed to the nucleus, the precession of the electron is identical to the effect of a magnetic field.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_T = \frac{1}{\gamma_e} \vec{\Omega}_T. }

Substituting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma_e = e/mc} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a= - e^2 r / mr^3} into <xr id="EQ_soitwo" /> gives

$B_{T}=-{\frac {1}{2}}{\frac {e\hbar }{mc}}{\frac {1}{r^{3}}}{\bf {L}}$

Hence the total effective magnetic field is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B^\prime = \frac{1}{2} \frac{e\hbar}{mc}\frac{1}{r^3} {\bf L} }

This gives rise to a total spin-orbit interaction

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{so} = - \vec{\mu}\cdot B^\prime =\frac{e^2 \hbar^2}{2m^2c^2}\frac{1}{r^3} {\bf S}\cdot {\bf L} }

The Darwin Term

Electrons exhibit "Zitterbewegung", fluctuations in position on the order of the Compton wavelength, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar /mc} . As a result, the effective Coulombic potential is not Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)} , but some suitable average ${{\overline {V}}(r)}$, where the average is over the characteristic distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hbar /mc} . To evaluate this, expand Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r )} about $r$ in terms of a displacement Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s } ,

$V(r+s)=V(r)+{\vec {\nabla }}V\cdot s+{\frac {1}{2}}\sum _{ij}s_{xi}s_{xj}{\frac {\partial ^{2}V}{\partial x_{i}\partial x_{j}}}+\cdots$

Assume that the fluctuations are isotropic. Then the time average of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r + s ) - V ( r)} is

$\Delta V\sim {\frac {1}{2}}{\left[{\frac {1}{3}}{\left({\frac {\hbar }{mc}}\right)^{2}}\right]}\nabla ^{2}V=-{\frac {1}{6}}{\frac {e^{2}\hbar ^{2}}{m^{2}c^{2}}}\nabla ^{2}\left({\frac {1}{r}}\right)$

The precise expression for the Darwin term is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm{Dar}} = -\frac{1}{8} \frac{e^2\hbar^2}{m^2c^2} \nabla^2 {\left(\frac{1}{r}\right)} }

The coefficient of the Darwin term is 1/8, rather than 1/6. Recall that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^2(\frac{1}{r})=-4\pi\delta^3(r)} , so that the Darwin term is nonzero only for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} states, where $\langle S|H_{\textrm {Dar}}|S\rangle >0$. That is, the Zitterbewegung effectively cuts off the cusp of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1/r} potential, thereby reducing the binding energy for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} -electrons.

Evaluation of the fine structure interaction

Fine structure of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=2} levels in hydrogen. The degeneracy between the $^{2}S_{1/2}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^2P_{1/2}} levels, which looks accidental in non-relativistic quantum mechanics, is really deeply rooted in the relativistic nature of the system. The degeneracy is ultimatly broken in QED by the Lamb Shift.

The spin orbit-interaction $H_{\rm {so}}$ is not diagonal in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} due to the term Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\cdot S} . However, it is diagonal in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bf {J} = \bf {L} + \bf {S}} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{\rm{Dar}}} is likewise diagonal in ${\bf {J}}$. Hence, finding the energy level structure due to the fine structure interaction involves evaluating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle < n, \ell, S, j, m_j | H_{\rm{FS}}| n, l, S, j, m_j >} . Note that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <H_{\rm{so}}>} vanishes in an $S$ state, and that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle <H_{\rm{Dar}}>} vanishes in all states but an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} state. Summing all the contributions, one finally obtains

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\rm{FS}} (n,j)=(\alpha^2 mc^2 ){\left( -\frac{\alpha^2}{2n^4}\right)} {\left( \frac{n}{j+1/2} -\frac{3}{4}\right)} }

Note that states of a given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and $j$ are degenerate. This degeneracy is a crucial feature of the Dirac theory.

The Lamb Shift

According to the Dirac theory, states of the hydrogen atom with the same values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} are degenerate. Hence, in a given term, (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^2S_{1/2} ,^2P_{1/2}} ), (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^2P_{3/2} , ^2D_{3/2}} ), (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^2D_{5/2} , ^2F_{5/2}} ), etc. form degenerate doublets. However, this is not exactly the case. Because of vacuum interactions, not taken into account in the Dirac theory, the degeneracy is broken. The largest effect is in the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2} state. The energy splitting between the $^{2}S_{1/2}$ and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ^2P_{1/2}} states is called the Lamb Shift.

This splitting is caused by interactions with the vacuum. A full treatment in QED (see books by Cohen-Tannoudji et al.) uses the operator for the interaction between the electromagnetic field and the atoms. In the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A p } representation, the Hamiltonian is $H_{I}=-(q/m)pA+(q^{2}/2m)A^{2}$. The operator of the vector potential A is the operator of the quantized radiation field. Carrying out second order perturbation theory (to order Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^2 } ) including summation over all modes gives the Lamb shift.

A simple physical model due to Welton and Weisskopf demonstrate its origin. In the following derivation, note the analogy between the vacuum fluctuations inducing the Lamb Shift and the Zitterbewegung responsible for the Darwin term treated in the previous section. Because of zero point fluctuation in the vacuum, empty space is not truly empty. The electromagnetic modes of free space behave like harmonic oscillators, each with zero-point energy Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h \nu/2} . The density of modes per unit frequency interval and per volume is given by the well known expression

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho (\nu ) d\nu = 8\pi \frac{\nu^2}{c^3}d\nu }

Consequently, the zero-point energy density is

$W_{\nu }={\frac {1}{2}}h\nu \rho (\nu )=4\pi {\frac {h\nu ^{3}}{c^{3}}}$

With this energy we can associate a spectral density of radiation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W_{\nu} = \frac{1}{8\pi} {\left( {\overline{E_{\nu}^2}} + {\overline{B_{\nu}^2}}\right)} = \frac{1}{8\pi}E_{\nu}^2 }

The bar denotes a time average and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_{\nu}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_{\nu}} are the field amplitudes. Hence,

$E_{\nu }^{2}={\frac {32\pi ^{2}h\nu ^{3}}{c^{3}}}$

For the moment we shall treat the electron as if it were free. Its motion is given by

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot{s}_{\nu} = eE_{\nu} \cos 2 \pi \nu t }

{EQ_lambsix}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\overline{s_{\nu}^2}} = \frac{e^2}{32 \pi^4 m^2\nu^4} E_{\nu}^2 ~~~ =\frac{e^2h}{\pi^2m^2c^3}\frac{1}{\nu} }

The effect of the fluctuation $s_{\nu }$ is to cause a change Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta V} in the average potential

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta V = {\overline{V(r+s_{\nu})}} -V(r) }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V ( r+s_{\nu})} can be found by a Taylor's expansion:

$V(r+s_{\nu })=V(r)+\Delta V\cdot {\bf {s}}_{\nu }+{\frac {1}{2}}\sum _{ij}{\frac {\partial ^{2}V}{\partial s_{{\nu },i}\partial s_{{\nu },j}}}s_{{\nu },i}s_{{\nu },j}+\cdots$

When we average this in time, the second term vanishes because s averages to zero. For the same reason, in the final term, only contributions with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = j} remain. We have, taking the average,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{V(r + s_{\nu})} = V(r) + \frac{1}{2} \sum_{i} \frac{\partial^2V_i}{\partial s_{{\nu},i}^2 } \overline{s_{{\nu},i}^{ 2}} }

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{s_{\nu ,i}^{2}} = \overline{s_{\nu}^{2}} /3} we obtain finally

${\overline {V(r+s_{\nu })}}={\frac {s_{\nu }^{2}}{6}}\nabla ^{2}V(r)$

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^2 V ( r) = 4 \pi Ze^{2}\delta ( r)} , we obtain the following expression for the change in energy

{EQ_lambeleven}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W_{\nu} = \frac{2\pi}{3}Z e^2s_{\nu}^2 < n^\prime , \ell^\prime , m^\prime |\delta (r) | n, \ell , m > }

The matrix element gives contributions only for $S$ states, where its value is

{EQ_lambtweleve}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle | \Psi_{n,0,0} (0) |^2 = \frac{Z^3}{\pi n^3 a_0^3} }

Combining Eqs. \ref{EQ_lambsix}, \ref{EQ_lambtweleve} into Eq. \ref{EQ_lambeleven} yields

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W_{\nu} = \frac{2}{3} e^2 s_{\nu}^2 \frac{Z^4}{n^3a_0^3} = \frac{2}{3} \frac{e^4Z^4}{m^2c^3\pi^2} \frac{1}{n^3a_0^3} \frac{h}{\nu} }

Integrating over some yet to be specified frequency limits, we obtain

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W = \frac{2}{3} \frac{e^4}{m^2c^3} \frac{Z^4}{\pi^2}\frac{h}{n^3a_0^3} \mbox{ln} {\left( \frac{\nu_{\rm max}}{\nu_{\rm min}}\right) } }

At this point, atomic units come in handy. Converting by the usual prescription, we obtain

$\delta W={\frac {4}{3\pi }}\alpha ^{3}{\frac {Z^{4}}{n^{3}}}{\mbox{ln}}{\left({\frac {\nu _{\rm {max}}}{\nu _{\rm {min}}}}\right)}\ {\textrm {hartree}}={\frac {4}{3\pi }}\alpha ^{5}mc^{2}{\frac {Z^{4}}{n^{3}}}{\mbox{ln}}{\left({\frac {\nu _{\rm {max}}}{\nu _{\rm {min}}}}\right)}$

The question remaining is how to choose the cut-off frequencies for the integration. It is reasonable that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nu_{\rm min}} is approximately the frequency of an orbiting electron, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z^2 /n^3} in atomic units. At lower energies, the electron could not respond. For the upper limit, a plausible guess is the rest energy of the electron, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m c^2} . Hence, $\nu _{\rm {max}}/\nu _{\rm {min}}\sim n^{3}/(Z^{2}\alpha ^{2})$. For the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2S} state, this gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta W = \frac{1}{6\pi} \alpha^3 \ln \frac{8}{\alpha^2} = 2.46 \times 10^{-7}} atomic~units Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 1,600} MHz

The actual value is 1,058 MHz. Measurements of the Lamb shift \cite{Lamb1947, Weitz1994, Schwob1999} have occupied the forefront of hydrogen spectroscopy from Lamb's original 1947 discovery, using microwave techniques, of a shift of "about 1000 Mc/sec" to more recent results at the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-7}} level of precision obtained in the optical domain by two-photon spectroscopy.

Discussion on the sign of the Lamb shift:

The physical picture due to Welton and Weisskopf implies that the Coulomb field is "smeared out" for s electrons and results in a weaker binding energy. This is correct. However, there is a second contribution, the vacuum polarization which can be visualized by assuming that between charged particles virtual electrons and positrons pairs are created which shield some of the Coulomb field. One would expect that this shielding effect lowers the biding energy. However, what we measure as charge of the proton and electron are the shielded values of the charge. An s electron interacting with the proton penetrates the shield and sees a charge which is higher, closer to the naked charge. Therefore, the vacuum polarization increases the binding energy of s electrons. For the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2S_{1/2}} state this amounts to 27 MHz, or 3% of the total Lamb shift.

References

• Lamb1947: Willis E. Lamb and Robert C. Retherford. Fine structure of the hydrogen atom by a microwave method. Phys. Rev., 72(3):241–243, Aug 1947.
• Schwob1999: C. Schwob, L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, J.-J. Zondy, and A. Clairon. Optical frequency measurement of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2s-12d} transitions in hydrogen and deuterium: Rydberg constant and Lamb shift determinations. Phys. Rev. Lett., 82(25):4960–4963, Jun 1999.
• Weitz1994: M. Weitz, A. Huber, F. Schmidt-Kaler, D. Leibfried, and T. W. Hänsch. Precision measurement of the hydrogen and deuterium 1 s ground state Lamb shift. Phys. Rev. Lett., 72(3):328–331, Jan 1994.