The electric dipole approximation

In general, the wave-length of the type of electromagnetic radiation which induces, or is emitted during, transitions between different atomic energy levels is much larger than the typical size of a light atom. Thus,

$$\exp[ {\rm i} (\omega/c) {\bf n}\!\cdot\!{\bf r}] = 1 + {\rm i} \frac{\omega}{c} {\bf n}\!\cdot\!{\bf r} + \cdots,$$ (845)

can be approximated by its first term, unity (remember that $\omega/c =2\pi/\lambda$). This approximation is known as the electric dipole approximation. It follows that

$$\langle n\vert \exp[ {\rm i} (\omega/c) {\bf n}\!\cdot\!{... ...dmath$\epsilon$}\!\cdot\! \langle n\vert{\bf p}\vert i\rangle.$$ (846)

It is readily demonstrated that

$$[{\bf r}, H_0]= \frac{{\rm i} \hbar {\bf p}}{m_e},$$ (847)

so

$$\langle n\vert {\bf p}\vert i\rangle = -{\rm i} \frac{m_e}... ...rm i} m_e \omega_{ni} \langle n\vert{\bf r}\vert i\rangle.$$ (848)

Using Eq. (844), we obtain

$$\sigma_{\rm abs} = 4\pi^2 \alpha \omega_{ni} \vert\langl... ...dot\!{\bf r}\vert i\rangle\vert^2 \delta(\omega-\omega_{ni}),$$ (849)

where $\alpha = e^2/(2\epsilon_0 h c) = 1/137$ is the fine structure constant. It is clear that if the absorption cross-section is regarded as a function of the applied frequency, $\omega$, then it exhibits a sharp maximum at $\omega = \omega_{ni} =(E_n - E_i)/\hbar$.

Suppose that the radiation is polarized in the $z$-direction, so that $\mbox{\boldmath$\epsilon$} = \hat{\bf z}$. We have already seen, from Sect. 6.4, that $\langle n\vert z\vert i\rangle=0$ unless the initial and final states satisfy

$$\Delta l = \pm 1,$$ (850)

$$\Delta m = 0.$$ (851)

Here, $l$ is the quantum number describing the total orbital angular momentum of the electron, and $m$ is the quantum number describing the projection of the orbital angular momentum along the $z$-axis. It is easily demonstrated that $\langle n\vert x\vert i \rangle$ and $\langle n\vert y\vert i\rangle$ are only non-zero if

$$\Delta l = \pm 1,$$ (852)

$$\Delta m = \pm 1.$$ (853)

Thus, for generally directed radiation $\langle n\vert\mbox{\boldmath$\epsilon$} \!\cdot\!{\bf r}\vert i\rangle$ is only non-zero if

$$\Delta l = \pm 1,$$ (854)

$$\Delta m = 0, \pm 1.$$ (855)

These are termed the selection rules for electric dipole transitions. It is clear, for instance, that the electric dipole approximation allows a transition from a $2p$ state to a $1s$ state, but disallows a transition from a $2s$ to a $1s$ state. The latter transition is called a forbidden transition.

Forbidden transitions are not strictly forbidden. Instead, they take place at a far lower rate than transitions which are allowed according to the electric dipole approximation. After electric dipole transitions, the next most likely type of transition is a magnetic dipole transition, which is due to the interaction between the electron spin and the oscillating magnetic field of the incident electromagnetic radiation. Magnetic dipole transitions are typically about $10^5$ times more unlikely than similar electric dipole transitions. The first-order term in Eq. (845) yields so-called electric quadrupole transitions. These are typically about $10^8$ times more unlikely than electric dipole transitions. Magnetic dipole and electric quadrupole transitions satisfy different selection rules than electric dipole transitions: for instance, the selection rules for electric quadrupole transitions are $\Delta l =0, \pm 2$. Thus, transitions which are forbidden as electric dipole transitions may well be allowed as magnetic dipole or electric quadrupole transitions.

Integrating Eq. (849) over all possible frequencies of the incident radiation yields

$$\int \sigma_{\rm abs} (\omega) d\omega = \sum_n 4\pi^2 \al... ...ox{\boldmath$\epsilon$} \!\cdot\!{\bf r}\vert i\rangle\vert^2.$$ (856)

Suppose, for the sake of definiteness, that the incident radiation is polarized in the $x$-direction. It is easily demonstrated that

$$[x, [x, H_0] ] = - \frac{\hbar^2}{m_e}.$$ (857)

Thus,

$$\langle i\vert[ x, [x, H_0] ] \vert i\rangle = \langle i\v... ..., x^2 - 2 x H_0 x\vert i\rangle = - \frac{\hbar^2}{m_e},$$ (858)

giving

$$2\sum_n \left(\langle i\vert x\vert n\rangle E_i \langle n\v... ...\langle n\vert x\vert i\rangle\right) = - \frac{\hbar^2}{m_e}.$$ (859)

It follows that

$$\frac{2 m_e}{\hbar} \sum_n \omega_{ni} \vert\langle n\vert x\vert i\rangle\vert^2 = 1.$$ (860)

This is known as the Thomas-Reiche-Kuhn sum rule. According to this rule, Eq. (856) reduces to

$$\int \sigma_{\rm abs} (\omega) d\omega = \frac{2\pi^2 \alpha \hbar}{m_e} = \frac{\pi e^2}{2 \epsilon_0 m_e c}.$$ (861)

Note that $\hbar$ has dropped out of the final result. In fact, the above formula is exactly the same as that obtained classically by treating the electron as an oscillator.