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Lamb Shift

Two contribution:

  • Coulomb potential is "smeared out", so a weaker binding energy.
  • Futher contribution to the Lamb shift. 27 MHz, 3% : vaccum polarization. We observe the shielded charge. $\Rightarrow$ s(l = 0) electron sees a higher charge(s 电子 穿透到里面) $\Rightarrow$ down shift for $2S_{1/2}$ 27MHz.

vaccum polarization

Introduction: Effect of Nucleus (Hyperfine Structure)

So far: point charge

Now: nucleus has structure

mangnetic moment: dipole or quadrapole $\Rightarrow$ HFS.

mass and volume $\Rightarrow$ Isotope shifts

hyperfine structure

Relevance of hpyerfine structure:

  • several ground states (optical pumping, manuplation of atoms)
  • astronomical observation 21cm line
  • determination of nuclear properties(even of unstable nuclei)


the most important effect due to the hyperfine structure: the nucleolus has a magnetic moment, and this magnetic moment couples to the magnetic field, even if you don't apply an external magnetic field.

$$\begin{align} H_{\mathrm{HF}} = - \vec{B}_J(0)\cdot \vec{\mu}_{I} \end{align}$$

this is the Zeeman Hamiltonian of the nucleus in the magnetic field created by the electron.

Alternatively: $e^-$ moves in the potential of the nucleus \begin{align} V = - \frac{e^2}{r} , \vec{A} = \frac{\vec{\mu}_I\times \mathrm{r}}{r^3} \quad \mathrm{Jackson(5.55)} \end{align} then solve the Schrodinger Eq.

Semi classical derivation of $\vec{B}_{J}$

orbital and spin part:

$$\begin{align} \vec{B}_J = \vec{B}_L + \vec{B}_S \end{align}$$

Jackson (5.64):

$$\begin{align} \vec{B}_S = \frac{1}{r^3}[-\vec{\mu} + 3(\vec{\mu}\cdot \hat{r})\hat{r}] + \frac{8\pi}{3}\delta(\vec{r}) \vec{\mu} \end{align}$$

where $\vec{\mu} = g\mu_B\vec{S}$. Biot and Savart:

$$\begin{align} \vec{B}_L(0) = \frac{1}{c}\int \vec{I} \frac{\mathrm{d}\vec{r}\times \vec{r}}{r^3} \rightarrow \frac{e}{c} \int \rho (\vec{r}) \frac{\vec{v}\times \vec{r}}{r^3} = - \frac{e}{mc}\vec{L} \langle \frac{1}{r^3} \rangle = - 2\mu_B \vec{L} \langle \frac{1}{r^3} \rangle \end{align}$$

total $\vec{B}$ field (using $g= -2$ for electron spin)

$$\begin{align} \vec{B}_J = -2\mu_B \left( \frac{\vec{L}}{r^3} - \frac{\vec{S}}{r^3} + \frac{3(\vec{S}\cdot \hat{r})\hat{r}}{r^{3}} + \frac{8}{3}\pi \delta(\vec{r})\vec{S} \right) \end{align}$$

the first three terms is $0$ for $L = 0$ because of spherical average, the last term is $0$ for $L\neq 0$ .

$$\begin{align} H_{HF} = - \vec{B}_J \vec{\mu}_I = a h \vec{I} \cdot \vec{J} \end{align}$$

For $L = 0$ , $\langle \vec{J} \cdot \vec{B}_J = - 4\pi \mu_B |\psi_{s}|^2$

$$\begin{align} a h = \frac{g_I \mu_N \mu_B 4\pi |\psi(0)|^2}{S(S + 1)} , \quad \vec{\mu}_I = g_I \mu_N \vec{I} \end{align}$$

For hydrogen: 1420MHz

$$\begin{align} H = a h \vec{I}\cdot\vec{J} = \frac{a h}{2} (F(F + 1) - I(I + 1) - J(J + 1)) \end{align}$$

$I = J = S = 1/2$, $F = 1, 0$

hyperfine structure for hydrogen

center of mass of HF states is not change.

Quadrapole moments

What further (electric or magnetic) moments can a nucleus have?

parity of electric multiple: $( - 1)^l$ parity parity of magnetic multiple: $( - 1)^{ l + 1}$ parity

Minimum I for magnetic dipole and electric quadrapole

for magnetic dipole is $I\ge 1/2$ : 分清两个方向, 只需要翻个个就行

for electric quadrapole is $I \ge 1$ : 分清一个椭球还是一个球, 需要有一个反转 90 度的操作


Isotopic effects


$$\begin{align} E_n = E_n^{\infty} (\frac{M}{M + m}) \end{align}$$

$M$ : nucleus mass. $m$ : electron mass


Coulomb level shits. only $s$ electron

weakens the binding energies

Volum effect: Largest for heavy nuclei

Mass effect: Largest for hightest nuclei