7.9: Appendix- Magnetic Dipole and Electric Quadrupole Transitions

The leading term in the expansion of \(e^{\mathrm{i}\mathbf{k} \cdot \mathbf{r}}\) in the light-matter interaction Hamiltonian, eq. (7.4.6), resulted in electric dipole transitions, also known as E1 transitions. The second term

\[V^{(2)}(t) = - \mathrm{i}\frac{q}{m}\tilde A_{0} ( \hat \varepsilon \cdot \mathbf{p} )( \mathbf{k} \cdot \mathbf{r} )e^{ - \mathrm{i}\omega t} \label{MDEQT01}\]

leads to magnetic dipole (M1) and electric quadrupole (E2) transitions, which we will describe here. To start, let’s look at the product \(\left( \hat \varepsilon \, \cdot \,\mathbf{p} \right)\left( \mathbf{k} \cdot \mathbf{r} \right)\), which is now a function of two quantum operators, \(\mathbf{p}\) and \(\mathbf{r}\). The first term selects the projection of the particle’s momentum onto the direction of the field polarization. The second describes how the time-dependent position of the charge is projected along the direction of field propagation. Therefore, this term accounts for the motion and position of the charged particle in two orthogonal directions. To look closer, let’s consider the case that we have a plane wave polarized in the \(x\) direction and propagating along \(z\). In that case this product is

\[ ( \hat \varepsilon \cdot \mathbf{p} ) ( \mathbf{k} \cdot \mathbf{r} ) \to zp_{x} \label{MDEQT02} \nonumber\]

(Remember: since the directions are orthogonal, the operators commute, and the order doesn’t matter). You may note that this type of product appears in the orbital angular momentum operator \(L_{y} = zp_{x} - xp_{z}\), and indeed angular momentum is the type of operator that is expected to influence a particle’s position and momentum in both \(x\) and \(z\) dimensions. We therefore expect that, in evaluating the matrix elements \(\langle k|zp_{x}| \ell \rangle\), there might be contributions from \(\langle k|L_{y} | \ell \rangle\) and a second factor. The trick to making this work is to write the operator in the form \(zp_{x}\)

\[zp_{x} = \frac{1}{2}\left( zp_{x} - xp_{z} \right) + \frac{1}{2}\left( zp_{x} + xp_{z} \right) \label{MDEQT03} \nonumber\]

This strategy results in a symmetrized and anti-symmetrized operators that will commute. The first term is what we expected, \({\textstyle{1 \over 2}}L_{y}\). The second term can be simplified using the expression for a particle in a central potential: \(\left[ H_{0},xz \right] = - \mathrm{i} \hbar (zp_{x} + xp_{z})/m\). Then the resulting matrix elements of \(V^{\left( 2 \right)}\) can be written as

\[\langle k|zp_{x} | \ell \rangle = \frac{1}{2}\langle k|L_{y} | \ell \rangle + \frac{\mathrm{i} m}{2\hbar }\langle k|xz | \ell \rangle \label{MDEQT04} \]

So, what do these terms indicate? the leading term describes the orbital angular momentum in \(y\)—the direction in which the magnetic field is polarized—and this can be expressed in terms of a magnetic dipole moment. The second term represents simultaneous charge displacements in the \(z\) and \(x\) directions, and this can be expressed as an electric quadrupole moment. With this insight, we can return to the original expression and work through the problem more generally.

First, we can rewrite the operator product in eq. (\ref{MDEQT04}) using the outer product \(\mathbf{pr}\)

\[ ( \hat \varepsilon \cdot \mathbf{p} ) ( \mathbf{k} \cdot \mathbf{r}) = \hat \varepsilon \cdot ( \mathbf{pr} ) \cdot \mathbf{k} \label{MDEQT05}\]

Written explicitly, this is

\[\hat \varepsilon \cdot ( \mathbf{pr} ) \cdot \mathbf{k} = \left( \begin{array}{*{20}{c}} \varepsilon _{x}&\varepsilon _{y}&\varepsilon _{z} \end{array} \right)\left( \begin{array}{*{20}{c}} p_{x}r_{x}&p_{x}r_{y}&p_{x}r_{z}\\ p_{y}r_{x}&p_{y}r_{y}&p_{y}r_{z}\\ p_{z}r_{x}&p_{z}r_{y}&p_{z}r_{z} \end{array} \right)\left( \begin{array}{*{20}{c}} k_{x}\\ k_{y}\\ k_{z} \end{array} \right) \label{MDEQT06}\]

Alternatively, this could be written \(\sum\nolimits_{n}^{m} \varepsilon _{m} ( p_{m}r_{n})k_{n}\), where the indices \(m,n = \left\{ x,y,z \right\}\). We can now rewrite the product of operators \(\mathbf{pr}\) as a sum of symmetrized and antisymmetrized terms:

\[\hat \varepsilon \cdot (\mathbf{pr}) \cdot \mathbf{k} = \frac{1}{2}\hat \varepsilon \cdot ( \mathbf{pr} - \mathbf{rp} ) \cdot \mathbf{k} + \frac{1}{2}\hat \varepsilon \cdot ( \mathbf{pr} + \mathbf{rp} ) \cdot \mathbf{k} \label{MDEQT07}\]

We now have two commuting terms which will allow us to separate \(V^{(2)}(t)\) into two contributions: \(V _{\text{mag}}^{(2)}\) gives rise to magnetic dipole transitions, and \(V_Q^{(2)}\) leads to electric quadrupole transitions:

\[V^{(2)}(t) = V _{\text{mag}}^{(2)}(t) + V_Q^{(2)}(t) \label{MDEQT08}\]

Let’s now discuss and simplify these two terms to make them more intuitive. Considering the leading term in (\ref{MDEQT07}), we can use the vector identity \(( A \times B ) \cdot ( C \times D ) = ( A \times C ) ( B \times D ) - ( A \times D ) ( B \times C )\) to write this expression in terms of the orbital angular momentum \(\mathbf{L}\)

\[\begin{aligned}
\frac{1}{2}\hat \varepsilon \cdot ( \mathbf{pr} - \mathbf{rp} ) \cdot \mathbf{k} & =\frac{1}{2}\left[ ( \hat \varepsilon \cdot \mathbf{p} )( \mathbf{r} \cdot \mathbf{k} ) - ( \hat \varepsilon \cdot \mathbf{r} )( \mathbf{p} \cdot \mathbf{k} ) \right] \\
& = \frac{1}{2}\left[ ( \mathbf{k} \cdot \hat \varepsilon ) \cdot ( \mathbf{r} \times \mathbf{p} ) \right] \\
& = \frac{1}{2} ( \mathbf{k} \times \hat{\varepsilon} ) \cdot \mathbf{L} \end{aligned} \label{MDEQT09}\]

Since the vector \(\mathbf{k} \times \hat \varepsilon\) describes the direction of the magnetic field \( \boldsymbol{\mathcal{B}} ​​​​​​\), and the amplitude of the magnetic field and vector potential are related by \(\tilde A_{0} = { {\mathcal{\tilde B}}_{0} / \mathrm{i}k }\), we can write

\[V _{\text{mag}}^{(2)}(t) = - \frac{e}{2m} \boldsymbol{\mathcal{B}} (t) \cdot \mathbf{L} \label{MDEQT10}\]

with \(\boldsymbol{\mathcal{B}}(t) = \hat b{\mathcal{\tilde B}}_{0} e^{ - \mathrm{i}\omega t}\). We are dealing with electron orbital angular momentum, so we have set \(q = e\). Using the definition of the Bohr magneton

\[\mu _{B} = \frac{e \hbar}{2m} \approx 9.274 \times 10^{ - 24} \, \text{J} \, \text{T}^{ - 1} \label{MDEQT11} \nonumber\]

and the magnetic dipole operator

\[ \boldsymbol{\mu} _{\text{mag}} = - \frac{e}{2m} \mathbf{L} = - \mu _{B}\frac{ \mathbf{L} }{\hbar } \label{MDEQT11.5} \nonumber\]

we can see that the magnetic dipole interaction with the electromagnetic field is

\[V _{\text{mag}}^{(2)}(t) = - \boldsymbol{\mathcal{B}}(t) \cdot \boldsymbol{\mu} _{\text{mag}} \label{MDEQT12} \]

This is the same form that we found for electric dipole interactions with the electric field.

Remember that we have not included electron spin \(\mathbf{S}\) in our derivations of the light-matter interactions. Including magnetic field interactions with electron spin in the original Hamiltonian leads to a more general form in which \(\boldsymbol{\mathcal{B}} \cdot \mathbf{L}\) in eq. (\ref{MDEQT10}) is replaced with \(\boldsymbol{\mathcal{B}} \cdot \left( \mathbf{L} + g_{e}\mathbf{S} \right)\), where the electron factor \(g\) is \(g_{e} \approx 2\). As a result, there are two contributions to the magnetic dipole moment: one from the electron orbital angular momentum and one from the electron spin angular momentum: \( \boldsymbol{\mu} _{\text{mag}} = \boldsymbol{\mu} _{L} + \boldsymbol{\mu }_{S}\).

Turning to electric quadrupole transitions, we can simplify the expression (\( \mathbf{pr} + \mathbf{rp} \)) in eq. (\ref{MDEQT07}) by substituting the identity \(\mathbf{p} = \mathrm{i}m\left[ H_{0},\mathbf{r} \right]/\hbar\):

\[\mathbf{pr} + \mathbf{rp} = \frac{\mathrm{i}m}{\hbar }\left[ \left[ H_{0},\mathbf{r} \right]\mathbf{r} - \mathbf{r}\left[ \mathbf{r},H_{0} \right] \right] = \frac{ - \mathrm{i}m}{\hbar }\left[ \mathbf{rr},H_{0} \right] \label{MDEQT13}\]

The resulting matrix element for this operator is \(\langle k|\left[ \mathbf{rr},H_{0} \right] | \ell \rangle = \hbar \omega _{k\ell }\langle k|\mathbf{rr} | \ell \rangle\). Here, \(\mathbf{rr}\) is an outer product of two vectors, which is proportional to the quadrupole moment. In a system with spherical symmetry, the dipole moment describes charge displacements introduce an asymmetry in the charge distribution, which generates a force or torque in an external electric field. Quadrupole moments involve changes in the charge displacement that introduce asymmetry with no net dipole and no net force on an external electric field. This quadrupole moment is obtained from \(\mathbf{Q} = q\mathbf{rr}\), which can be written as a second rank tensor (\(3 \times 3\) matrix) \(Q_{m,n} = \sum _{m,n}qr_{m}r_{n}\), where \(m,n = \left\{ x,y,z \right\}\). Motivated by the classical definition, with the recognition that \(\mathbf{Q}\) is a traceless matrix (\(\sum_mQ_{mm}=0\)), we define the quadrupole moment for many charges (\(i\)) as

\[Q_{m,n} = \sum_{i} q_{i}\left( 3r_{m,i} \cdot r_{n,i} - r_i^{2}\delta _{m,n} \right) \quad \quad m,n = \{ x,y,z\} \label{MDEQT14}\]

We can now express the interaction potential of the quadrupole moment with an electric field using \(\tilde A_{0} = { {\tilde E}_{0} / \mathrm{i} \omega }\)

\[\begin{aligned}
V_Q^{(2)} (t) & =- \mathrm{i}\frac{q}{2m} \tilde{\mathbf{A}}_{0} \cdot \left( \mathbf{pr} + \mathbf{rp} \right) \cdot \mathbf{k} e^{ - \mathrm{i}\omega t} \\
&= - \frac{1}{2\mathrm{i}\hbar \omega } \tilde{\mathbf{E}}_{0} \cdot \left[ \mathbf{Q},H_{0} \right] \cdot \mathbf{k} e^{ - \mathrm{i}\omega t} \\
\end{aligned} \label{MDEQT15} \]

Then, the transition quadrupole matrix element can be evaluated as

\[\begin{aligned}
\left\langle k \left|V_Q^{(2)}\right| \ell \right\rangle &= \frac{\mathrm{i} \tilde{E}_{0}\omega _{k\ell }}{2\omega }\left\langle k \right|\hat \varepsilon \cdot \mathbf{Q} \cdot \hat k\left| \ell \right\rangle \\
&= \frac{\mathrm{i} \tilde{E}_{0}\omega _{k\ell }}{2\omega } \mathbf{Q}_{k,\ell } \end{aligned} \label{MDEQT16}\]

What are the selection rules for these light-matter interaction mechanisms? Let’s consider the example of atomic spectroscopy. The selection rules for electric-dipole (or E1) transitions between states \(k\) and \(\ell\) dictates that states \(\ell\) and \(k\) have to be of opposite parity and that the changes in angular momentum quantum numbers are \(\Delta\ell=\pm1\) and \(\Delta m_\ell=0,\pm1\). For magnetic dipole (M1) and electric quadrupole (E2) transitions, states \(\ell\) and \(k\) have to be of the same parity, and the allowed changes in quantum numbers are \(\Delta\ell=\pm1\) and \(\Delta m_\ell=0,\pm1\).

---