Pre-Quantum Electrodynamics

• PM 10.4, 10.5
• Gr 4.1

Let us briefly discuss how electric fields affect (charged) matter.

Effect on fundamental particles: admittedly a complicated business, for which quantum mechanics is required. The assumption we make is that the charges of the particles we're dealing with are never changed by anything that happens to them. This is however not true in the real world, because the charges we measure are usually the remnants of some complicated intermingling of constituent particles. For example, the measured charge of an electron is a renormalized charge (it depends on how closely you probe the electron, in other words at what collision/scattering energies you probe it). This goes much beyond the current course, and needs the whole machinery of quantum electrodynamics.

For us here, we will assume that fundamental particles remain 'unchanged' in the presence of an E field, irrespective of how strong the latter is.

Simple atoms: for example Hydrogen. We simply have a nucleus (proton) with an orbiting electron. The mass ratio between these is about 1800 to 1. To treat this, we'd need to start from the nonrelativistic Schrödinger equation for the electron (assuming we're in the center-of-mass frame) in the presence of a constant (for simplicity) external electric field \({\bf E}\): \[ -\frac{\hbar^2}{2m} {\boldsymbol \nabla}^2 \psi - \frac{e^2}{4\pi \varepsilon_0 r} \psi - e {\bf E} \cdot {\bf r} ~\psi = E \psi. \] As compared to the zero-field case, the energy levels are modified changed (by the Stark effect, linear and nonlinear (latter for case of hydrogen in fundamental level); see Landau Lifschitz, vol 3 nr 77).

If the field is small, one can use perturbation theory. This gives an electric dipole moment of \[ {\bf p} = \langle \psi | (-e {\bf r}) | \psi \rangle = ... = \frac{9}{2} (4\pi \varepsilon_0 a_B^3) {\bf E} \] where \(a_B\) is the Bohr radius, \(a_B = \hbar^2/m e^2\).

Although the numerical factor is not guessable, the overall form is: Le Châtelier's principle tells us that the equilibrium position moves linearly with the strength of the perburtation.

More complex atoms: we face a similar scenario. The nucleus is now relatively even heavier than each electron. At small fields, we can neglect nonlinear effects (e.g. a given electron orbital change leading to changes in other orbitals). We still expect to have some induced dipole moment which increases linearly with the external field, \[ {\bf p} = \alpha {\bf E} \] except that now we have to solve a much more complicated QM problem. The factor \(\alpha\) is an atom-specific number called the atomic polarizability.

Molecules: atoms can now 'share' electrons, so the charge distribution can become nontrivial. Example: carbon dioxide, \(O - C - O\). Higher polarizability along axis than perpendicular to axis. In totally non-symmetric case: expect \[ p_i = \sum_{j = x,y,z} \alpha_{ij} E_j \] where \(\alpha_{ij}\) is the polarizability tensor of the molecule. Always possible to use 'principal' axes such that all but 3 of the terms cancel.

Polar molecules: unlike individual atoms, molecules can have a permanent dipole moment. These are called polar molecules. Example: \(H Cl\) has elecronic density more closely bound on \(Cl\) than \(H\), so has a dipole moment pointing from \(Cl\) to \(H\). Other example: water, with \(105^\circ\) angle between the \(H^+\) and \(O^-\), dipole moment pointing from \(O^-\) along bisector.

Torque on dipole: if field is uniform, overall force on dipole cancels, but torque remains:

If field is non-uniform, \[ {\bf F} = {\bf F}_+ + {\bf F}_- = q({\bf E}_+ - {\bf E}_-) = q ({\bf d} \cdot {\boldsymbol \nabla}) {\bf E} \] so

The energy of a dipole in an electric field can be shown to be

\[ U = -{\bf p} \cdot {\bf E} \tag{di_E}\label{di_E} \]

Many atoms: gases and liquids: here, atoms or molecules are still more or less free from each other's influence as far as polarization is concerned. Random thermal motion, external field gives preferential direction to polarization. The relationship is still linear.

Many atoms: solids: here, things can be more complicated. Material can be insulating or conducting. If conducting: external field makes charges move such that interior becomes equipotential. If insulating: each constituent atom/molecule can pick up an induced polarization, polar molecules can tend to line up, crystal structure can be deformed, …

For zero field, the solid can have either zero or nonzero polarization. If nonzero: we call this spontaneous polarization, or rather ferroelectricity (after ferromagnetism, which is the correspdonding magnetic phenomenon happening with iron).