Electric Dipole Transitions

In general, the wavelength of the type of electromagnetic radiation that induces, or is emitted during, transitions between different atomic states is much larger than the typical size of a light atom. Thus, recalling that $\omega/c =2\pi/\lambda$ , the expession

$$\exp\left[\,{\rm i}\left(\frac{\omega}{c}\right){\bf n}\cdot{\bf x}\right] = 1 + {\rm i}\,\frac{\omega}{c} \,{\bf n}\cdot{\bf x} + \cdots,$$ (8.164)

is well approximated by its first term, unity. This approximation is known as the electric dipole approximation. It follows from Equation (8.136) that

$${\bf d}_{if} \simeq \frac{-{\rm i}}{m_e\,\omega_{if}}\, \langle i\vert\,{\bf p}\,\vert f\rangle.$$ (8.165)

It is readily demonstrated from Equations (3.33) and (8.132) that

$$[{\bf x}, H_0] = \frac{{\rm i}\, \hbar \,{\bf p}}{m_e},$$ (8.166)

so

$$\langle i\vert\, {\bf p}\,\vert f\rangle = -{\rm i}\, \frac{m_e}{... ...f\rangle = {\rm i}\,m_e\,\omega_{if}\, \langle i\vert\,{\bf x}\,\vert f\rangle,$$ (8.167)

which implies that

$${\bf d}_{if} \simeq \langle i\vert\,{\bf x}\,\vert f\rangle.$$ (8.168)

Here, ${\bf d}_{if}$ is termed the electric dipole matrix element. Recall from Equations (8.149), (8.150), and (8.164) that

$$w_{i\rightarrow f}^{\,\rm abs} = 4\pi^2\,\alpha\,\frac{c}{\hbar}\... ...ga_{fi})\left\vert\mbox{\boldmath$\epsilon$}\cdot {\bf d}_{fi}\right\vert^{\,2}$$ (8.169)

for absorption, and

$$w_{i\rightarrow f}^{\,\rm stm} = 4\pi^2\,\alpha\,\frac{c}{\hbar}\... ...ga_{if})\left\vert\mbox{\boldmath$\epsilon$}\cdot {\bf d}_{if}\right\vert^{\,2}$$ (8.170)

for stimulated emission, and with

$$w_{i\rightarrow f}^{\,\rm spn} = \frac{4\,\alpha\,\omega_{if}^{\,3}\,\vert d_{if}\vert^{\,2}}{3\,c^{\,2}}$$ (8.171)

for spontaneous emission.

We have already seen, from Section 7.4, that $\langle i\vert\,z\,\vert f\rangle=0$ for a hydrogen-like atom, unless the initial and final states satisfy ${\mit\Delta} l = \pm 1$ and ${\mit\Delta} m = 0$ . Here, $l$ is the azimuthal quantum number, and $m$ the magnetic quantum number. Moreover, ${\mit\Delta}l$ is the difference between the azimuthal quantum numbers of the initial and final states, et cetera. It is easily demonstrated that $\langle i\vert\,x\,\vert f \rangle$ and $\langle i\vert\,y\,\vert f\rangle$ are only non-zero if ${\mit\Delta} l = \pm 1$ , and ${\mit\Delta} m = \pm 1$ . (See Exercise 10.) It follows that the electric dipole matrix elements, ${\bf d}_{if}= \langle i\vert\,{\bf x}\,\vert f\rangle$ , which control the rates of so-called electric dipole transitions, via Equations (8.172)-(8.174), are only non-zero if

$${\mit\Delta} l = \pm 1,$$ (8.172)

$${\mit\Delta} m = 0, \pm 1,$$ (8.173)

$${\mit\Delta} m_s = 0.$$ (8.174)

Here, $m_s$ is the spin quantum number, which is defined as the eigenvalue of $S_z$ divided by $\hbar$ . (Of course, ${\mit\Delta}m_s=0$ because ${\bf x}$ does not explicitly depend on spin.) These expressions are termed the selection rules for electric dipole transitions. It is clear, for instance, that the electric dipole selection rules permit a transition from a $2p$ state to a $1s$ state of a hydrogen-like atom, but disallow a transition from a $2s$ to a $1s$ state. The latter transition is called a forbidden transition. The previous selection rules can also be written in the slightly more general form

$${\mit\Delta} j = 0, \pm 1,$$ (8.175)

$${\mit\Delta} m_j = 0, \pm 1,$$ (8.176)

where $j$ and $m_j$ are the standard quantum numbers associated with the total angular momentum, and the projection of the total angular momentum along the $z$ -axis, respectively [23]. (These new rules are evident from a perusal of the results of Exercise 15.) Note, however, that an electric dipole transition between two $j=0$ states is forbidden [23].

Let us estimate the typical spontaneous emission rate for an electric dipole transition in a hydrogen atom. We expect the matrix element ${\bf d}_{if}$ , defined in Equation (8.171), to be of order $a_0$ , where $a_0$ is the Bohr radius. We also expect $\omega_{if}= (E_i-E_f)/\hbar$ to be of order $\vert E_0\vert/\hbar$ , where $E_0$ is the hydrogen ground-state energy. It thus follows from Equation (8.174) that

$$w_{i\rightarrow f}^{\,\rm spn} \sim \alpha^{\,3}\,\omega_{if}\sim \alpha^{\,5}\,\frac{m_e\,c^{\,2}}{\hbar},$$ (8.177)

where $\alpha\simeq 1/137$ is the fine structure constant. This is an important result, because our perturbation expansion is based on the assumption that the transition rate between different energy eigenstates is much less than the frequency of phase oscillation of these states. In other words, $w_{i\rightarrow f}^{\rm\,spn} \ll \omega_{if}$ . This is indeed the case.

According to Equation (8.152), the absorption cross-section associated with atomic transitions between an initial state of energy $E_i$ and a final state of energy $E_f>E_i$ can be written

$$\sigma^{\,\rm abs}_{i\rightarrow f}(\omega)= 4\pi^2\,\alpha \,\om... ...ldmath$\epsilon$}\cdot {\bf d}_{fi}\right\vert^{\,2}\delta(\omega-\omega_{fi}),$$ (8.178)

where $\omega_{fi}=(E_f-E_i)/\hbar$ . Suppose, for the sake of argument, that the electromagnetic radiation is polarized in the $x$ -direction, so that $\epsilon$ $={\bf e}_x$ . According to Equation (8.181), if such radiation is incident on a hydrogen-like atom then the net absorption cross-section, summed over all final states, and integrated over all possible angular frequencies of the radiation, can be written

$$\int_{-\infty}^\infty d\omega \sum_f\sigma^{\,\rm abs}_{i\rightarrow f}(\omega) = 2\pi^{\,2}\,\alpha\,\frac{\hbar}{m_e}\sum_{f} F_{if},$$ (8.179)

where the dimensionless parameter

$$F_{if} = \frac{2\,m_e\,\omega_{fi}}{\hbar}\,\vert\langle i\vert\,x\,\vert f\rangle\vert^{\,2}$$ (8.180)

is termed the oscillator strength associated with radiation-induced electric dipole transitions between states $i$ and $f$ . Note that if state $i$ is the ground state (i.e., the lowest energy state) then the sum in Equation (8.182) is over all atomic states. On the other hand, if state $i$ is not the ground state then the sum is restricted to states whose energies are greater than $E_i$ (because we must have $E_f>E_i$ for absorption). Now, a straightforward generalization of the result proved in Exercise 6 yields the so-called Thomas-Reiche-Kuhn sum rule [110,70,90]:

$$\sum_f F_{if} =1,$$ (8.181)

where the sum is over all atomic states. (See Exercise 11.) Thus, it follows that, provided state $i$ is the ground state,

$$\int_{-\infty}^\infty d\omega \sum_f\sigma^{\,\rm abs}_{i\rightarrow f}(\omega) = 2\pi^{\,2}\,\alpha\,\frac{\hbar}{m_e}=2\pi^{\,2}\,r_e\,c,$$ (8.182)

where $r_e = e^{\,2}/(4\pi\,\epsilon_0\,m_e\,c^{\,2})$ is the classical electron radius. Note that the integrated absorption cross-section is independent of Planck's constant. In fact, the previous result can also be obtained from classical physics [67].