Fine Structure and Lamb Shift

Fine Structure and Lamb Shift

Fine Structure

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 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} 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 {\it kinetic, spin-orbit,} and {\it Darwin} terms, 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{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 $H_{\rm {Dar}}$, 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

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{kin}} =\frac{1}{8}\frac{p^4}{m^3c^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 ${\vec {\mu }}_{2}=-g_{e}\mu _{0}S$. The electron g-factor, 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 g_e = 2} . 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, {\it Classical Electrodynamics}). The rate of rotation, the Thomas precession, 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 \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.

$B_{T}={\frac {1}{\gamma _{e}}}{\vec {\Omega }}_{T}.$

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 $a=-e^{2}r/mr^{3}$ into Eq.\ \ref{EQ_soitwo} 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 B_T = -\frac{1}{2} \frac{e\hbar}{mc}\frac{1}{r^3} 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

$H_{so}=-{\vec {\mu }}\cdot B^{\prime }={\frac {e^{2}\hbar ^{2}}{2m^{2}c^{2}}}{\frac {1}{r^{3}}}{\bf {S}}\cdot L$

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 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 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 } ,

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)+ \vec{\nabla}V \cdot s + \frac{1}{2} \sum_{ij} s _{xi} s_{xj} \frac{\partial^2V}{\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

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 \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^2c^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 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 \langle S|H_{\textrm{Dar}}|S\rangle>0} . That is, the Zitterbewegung effectively cuts off the cusp of the $1/r$ 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

\begin{figure}

(caption: 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 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}} and $^{2}P_{1/2}$ 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.)

\end{figure} The spin orbit-interaction 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 {\bf 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}} . $H_{\rm {so}}$ and $H_{\rm {Dar}}$ are likewise 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}} . 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 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, 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 $S$ state. It is left as an exercise to show 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 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 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. 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, as described in Chapter 1, 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 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}} 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. 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 section \ref{SEC_dt}. 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 $h\nu /2$. 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

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}{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 $E_{\nu }$ 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,

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}^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 }

${\overline {s_{\nu }^{2}}}={\frac {e^{2}}{32\pi ^{4}m^{2}\nu ^{4}}}E_{\nu }^{2}~~~={\frac {e^{2}h}{\pi ^{2}m^{2}c^{3}}}{\frac {1}{\nu }}$

The effect of the fluctuation 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_{\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 {\bf 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\delta ( r)} , we obtain the following expression for the change in 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 \delta W_{\nu} = \frac{2\pi}{2} e^2s_{\nu}^2 < n^\prime , \ell^\prime , m^\prime |\delta (r) | n, \ell , m > }

The matrix element gives contributions 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 its value is

$|\Psi _{n,0,0}(0)|^{2}={\frac {Z^{3}}{\pi n^{3}a_{0}^{3}}}$

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_0^2} \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

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{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 $\nu _{\rm {min}}$ 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, 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 max} /\nu_{\rm min} \sim Z^2 /(n^3 \alpha^2 )} . For the $2S$ 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 \mbox{ln} \frac{8}{\alpha^2} = 2.46 \times 10^{-7}~\mbox{atomic~units}= 1,600 {\rm ~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. \putbib