Quantum Description of the Matter-Radiation Interaction

Abstract

Quantum physics describes the radiation field, matter systems (like atoms, molecules and solids) and their interactions. The processes of interaction consist of absorption, emission and diffusion of photons, the basis of many experimental techniques to study materials and their electronic properties. The states of the electron systems can be studied in a single-particle picture or in a many-electron scheme; the two approaches will be compared in the study of the elementary processes of interaction with light at different levels of approximation for the initial, final and intermediate states of the matter system.

Appendix

Constants and units. Consistently with the majority of textbooks of theoretical physics, the cgs unit systems is used here, in particular the non-razionalized system. In the alternative choice (razionalized system) the factors $4\pi$ would disappear in (5.1) and the electronic charge and the values of $\mathbf {B}$ and $\mathbf {E}$ would be divided by $\sqrt{4\pi }$.

Here $e =4.80324\, 10^{-10} e.s.u.$, with $e.s.u.\, =\,$(erg  cm)$^{\frac{1}{2}}$. The fields $\mathbf {E}$ and $\mathbf {B}$ are measured in the same units.

The fine structure constant is a number whose value is about $(137,036)^{-1}$ in any system. It is written as

$$\begin{aligned} \alpha =\frac{e^{2}}{\hbar c} \end{aligned}$$

(5.48)

The value of the classical electron radius $r_{0}$, which appears in some expressions, is defined by the equality:

$$\begin{aligned} \frac{e^{2}}{r_{0}^{{}}}=mc^{2} \end{aligned}$$

where $m$ is the electron mass; it is

$$\begin{aligned} r_{0}=\frac{e^{2}}{mc^{2}}=2.81794\;10^{-13}\,\mathrm{{cm}} \end{aligned}$$

(5.49)

This has nothing to do with the physical size of the electron. At distance much shorter of this value the electrodynamics is still valid and at length scales much lower the electron is still behaves like a pointlike elementary particle.

Mathematical notations. In this chapter from the first section we have used some symbols, whose definitions follow. The gradient operator $\nabla$ in three dimensions is

$$\begin{aligned} \nabla \equiv ( \frac{\partial }{\partial x},\frac{\partial }{\partial y},\frac{\partial }{\partial z}) \end{aligned}$$

(5.50)

We summarize in the following lines its actions on a a function $f(\mathbf {r})$ or on a vector $\mathbf {a}(\mathbf {r})$.

$$\begin{aligned} \nabla f = (\frac{\partial f}{\partial x},\frac{\partial f}{\partial y},\frac{\partial f}{\partial z}) \end{aligned}$$

(5.51)

is a vector also labeled grad $f$;

$$\begin{aligned} \nabla \cdot \mathbf {a} = \frac{\partial a_x}{\partial x}+\frac{\partial a_y}{\partial y}+\frac{\partial a_z}{\partial z} \end{aligned}$$

(5.52)

is a scalar (a function) also named div $\mathbf {a}$; while

$$\begin{aligned} \nabla \times \mathbf {a} = (\frac{\partial a_z}{\partial y}-\frac{\partial a_y}{\partial z}\,,\,\frac{\partial a_x}{\partial z}-\frac{\partial a_z}{\partial x}\,,\,\frac{\partial a_y}{\partial x}-\frac{\partial a_x}{\partial y}) \end{aligned}$$

(5.53)

is a vector, named rot $\mathbf {a}$.

Coherent states. The coherent states $\mid \varPhi _c\rangle$ mentioned at the end of Sect. 5.2, can give expectation value of the field similar to a classical wave. Here some more details on them are given. Restricting to a single mode $(\mathbf {k},r)$ the state can be expliciltly written as

$$\begin{aligned} \mid c \rangle =e^{-\frac{1}{2}\mid c \mid ^2}\sum _{n=0}^\infty \frac{c^n}{\sqrt{n!}}\mid n \rangle \;\;\;\mathrm { where}\;\;\;c=\mid c \mid e^{i\delta } \end{aligned}$$

(5.54)

is a complex number. This state is normalized, being $\langle c\mid c\rangle =1$ and it is an eigenstate of $a \equiv a_{\mathbf {k},r}$, as can be easily verified.

$$\begin{aligned} a\mid c \rangle =c\mid c \rangle . \end{aligned}$$

(5.55)

The eigenvalue $c$ (note that $a$ is non hermitian operator) can be any complex number. The squared modulus and the phase of $c$ are linked to to the average number of photons in the state, proportional to the square of the amplitude, and to the phase of the classical wave. By using the (5.22)–(5.24) one obtains evaluating the expectation value of the operator $n=a^\dagger a$ and of $(\varDelta n)^2 =n^2-\langle n\rangle ^2$ over the state $\mid c \rangle$,

$$\begin{aligned} \langle n\rangle =\langle c\mid n \mid c \rangle =\mid c \mid ^2= \overline{n}\;\;\;\;\;\; (\varDelta n)^2=\,\mid c \mid ^2 \end{aligned}$$

(5.56)

and then

$$\begin{aligned} \frac{\varDelta n}{\langle n\rangle }=\frac{1}{\langle n\rangle ^\frac{1}{2}}. \end{aligned}$$

(5.57)

As a consequence if $\overline{n}=\langle n\rangle =\mid c\mid ^2$ increases the standard deviation $\varDelta n$, its incertitude, becomes smaller than the average number of photons in the state. The expectation value of the field becomes

$$\begin{aligned} \langle c\mid \mathbf {E} \mid c\rangle =-{\varvec{\varepsilon }}_{\mathbf {k},s} 2 \left( \frac{\hbar \omega _\mathbf {k}}{2V}\right) ^\frac{1}{2} \mid c \mid \sin \left( \mathbf {k} \cdot \mathbf {r} - \omega _\mathbf {k} t +\delta \right) \end{aligned}$$

(5.58)

If $\overline{n}$ is high the field takes the form of a classical oscillating wave, as the uncertainity $(\triangle \mathbf {E})^2$ rimains limited. See [2].

Further terms in the interaction. The description given by (5.29) or (5.31) and (5.30) are appropriate for a system of scalar non-relativistic particles. In the most cases to describe the electrons in atoms and molecules, we must take into account the spin 1/2 of the electrons and the interactions involving spin. New terms can be obtained starting from the the Dirac equation, with a method (Foldy-Wouthuysen transformation) [3].

The relativistic corrections to $\fancyscript{H}_{S}^{0}$ for the system of electrons are

$$\begin{aligned} \frac{1}{2m^{2}c^{2}}(-e)\sum \limits _{i}\mathbf {S}_{i}\cdot \left( \mathbf {\nabla }\varphi (\mathbf {r}_{i})\times \mathbf {p}_{i}\right) -\frac{1}{8m^{3}c^{6}}\sum \limits _{i}\mathbf {p}_{i}^{4}-\frac{e\hbar ^{2}}{8m^{2}c^{2}}\sum \limits _{i} \nabla ^2 \varphi (\mathbf {r}_{i}) \end{aligned}$$

(5.59)

where the first term is the spin-orbit term, the second the correction to kinetic energy and the third the Darwin term. $\varphi (\mathbf {r})$ is the electric scalar potential seen by each electron in the system.

The presence of spin introduces two additional terms also in the interaction Hamiltonian $\fancyscript{H}_{I}$ : $\fancyscript{H}_{I}^{(3)}$ and $\fancyscript{H}_{I}^{(4)}$ to be added to (5.30)

$$\begin{aligned} \fancyscript{H}_{I}^{(3)}+\fancyscript{H}_{I}^{(4)}\equiv \frac{e}{mc}\sum _{i}\mathbf {S}_{i}\cdot \left( \mathbf {\nabla \times A(r}_{i},t\mathbf {)}\right) \mathbf {-}\frac{e^{2}}{2m^{2}c^{4}}\sum \limits _{i}\mathbf {S}_{i}\cdot \mathbf {((}\frac{\partial }{\partial t}\mathbf {A(r}_{i},t\mathbf {))\times A(r}_{i},t\mathbf {))} \end{aligned}$$

(5.60)

The first term describes the interaction of the intrinsic magnetic moments of the electrons with the field. The term $\fancyscript{H}_{I}^{(3)}$ adds, in the matrix element evaluation, the contribution of the intrinsic magnetic moment to the magnetic dipole (5.43) linked to the orbital momentum, that is already included in (5.30).

In (5.60) $\mathbf {S}_i$ is the intrinsic angular momentum vector operator for each electron, in $\hbar$ units

$$\begin{aligned} \mathbf {S}_i=\frac{1}{2}\mathbf {\sigma }_i \end{aligned}$$

(5.61)

where $\mathbf {\sigma }$ is a vector operator represented by the three Pauli matrices

$$\begin{aligned} \mathbf {\sigma } = (\sigma _1,\sigma _2,\sigma _3) = \left[ \left( \begin{array}{cc} 0 &{} \;1\\ 1 &{}\; 0 \\ \end{array}\right) , \left( \begin{array}{cc} 0 &{} -i\\ i &{} \;0 \\ \end{array}\right) , \left( \begin{array}{cc} 1 &{} \;0\\ 0 &{} -1\\ \end{array}\right) \right] . \end{aligned}$$

(5.62)