# (p.236) C Scattering processes

# (p.236) C Scattering processes

# C.1 Classical mechanics

## C.1.1 Kinematics of binary collisions

Consider classical particles that interact through pairwise conservative forces that decay sufficiently rapidly at large distances. We are interested in describing the collision of a pair of particles that come from infinity with velocities ${\text{c}}_{1}$ and ${\text{c}}_{2}$. By energy conservation it is not possible for the particles to become captured in bound states. Therefore, the interaction lasts for a finite time, after which the particles escape again to infinity with velocities ${\text{c}}_{1}^{\prime}$ and ${\text{c}}_{2}^{\prime}$. For simplicity the case of particles of equal mass is considered; the extension to dissimilar masses is direct. Energy and momentum conservation imply that

Defining the pre- and postcollisional relative velocities as $\text{g}={\text{c}}_{2}-{\text{c}}_{1}$ and ${\text{g}}^{\prime}={\text{c}}_{2}^{\prime}-{\text{c}}_{1}^{\prime}$, respectively, the general solution of eqns (C.1) is^{[}^{1}^{]}

where $\stackrel{\u02c6}{\mathbf{n}}$ is a unit vector that will be determined later in terms of the interparticle potential and the relative velocity, but up to now is arbitrary. If the particles interact through a central potential, $\stackrel{\u02c6}{\mathbf{n}}$ points to the centre of mass. Noting that ${\text{g}}^{\prime}\cdot \stackrel{\u02c6}{\mathbf{n}}=-\text{g}\cdot \stackrel{\u02c6}{\mathbf{n}}$, one can directly invert the relation (C.2) as

where we note that the same expression is recovered with the roles of $\text{g}$ and ${\text{g}}^{\prime}$ exchanged.

Following (Cercignani, 2013), the transformations $\text{g}\to {\text{g}}^{\prime}$ [eqn (C.2)] and ${\text{g}}^{\prime}\to \text{g}$ [eqn (C.3)] are characterised by the same Jacobian ${J}_{\stackrel{\u02c6}{\mathbf{n}}}$, which depends on the unit vector. The Jacobian of the combined transformation is hence ${J}_{\stackrel{\u02c6}{\mathbf{n}}}^{2}$. However, as the combined transformation yields the initial relative velocity, the Jacobian must be one, from which we obtain that $|{J}_{\stackrel{\u02c6}{\mathbf{n}}}|=1$. Finally, using that the centre-of-mass velocity is conserved, we get that the Jacobian of the transformation from the initial to the final velocities is one; that is,

an expression that is used in the derivation of the Lorentz and Boltzmann equations in Chapters 3 and 4.

## (p.237) C.1.2 Geometrical parameterisation

The unit vector $\stackrel{\u02c6}{\mathbf{n}}$ is determined by the geometry of the collision. In the centre-of-mass reference frame, the impact parameter *b* is defined as the distance from the projected unscattered trajectory to the origin. By conservation of energy, $|{\text{g}}^{\prime}|=|\text{g}|$, therefore angular momentum conservation implies that, for the postcollisional state, the impact parameter is also *b*. The deflection angle $\mathrm{\chi}$ is related to ${\mathit{\varphi}}_{0}$, which points to the point of minimum distance, by $\mathrm{\chi}=|\mathit{\pi}-2{\mathit{\varphi}}_{0}|$ (Fig. C.1). Furthermore, ${\mathit{\varphi}}_{0}$ is obtained by integrating the motion under the influence of the interparticle potential *U* (Landau and Lifshitz, 1976),

where $\mathit{\mu}={m}_{1}{m}_{2}/({m}_{1}+{m}_{2})$ is the reduced mass for particles of masses ${m}_{1}$ and ${m}_{2}$, which equals $\mathit{\mu}=m/2$ for particles of equal mass.

If there is a beam of particles that approach the centre, it is useful to define the differential cross section $\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathit{\sigma}$ as the number of particles that are scattered between $\mathrm{\chi}$ and $\mathrm{\chi}+\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathrm{\chi}$ divided by the particle flux *n*, which is assumed to be uniform. If there is a one-to-one relation between *b* and $\mathrm{\chi}$, the particles that will be deflected in the referred range of angles have impact parameters between $b(\mathrm{\chi})$ and $b(\mathrm{\chi})+\phantom{\rule{thinmathspace}{0ex}}\text{d}b(\mathrm{\chi})$. Now, by simple geometry, $\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathit{\sigma}=2\mathit{\pi}b\phantom{\rule{thinmathspace}{0ex}}\text{d}b$, which using the chain rule gives $\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathit{\sigma}=2\mathit{\pi}b(\mathrm{\chi})\left|\frac{\mathrm{\partial}b(\mathrm{\chi})}{\mathrm{\partial}\mathrm{\chi}}\right|\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathrm{\chi}$. Finally, instead of the differential of the deflection angle, the cross section is usually referred to the element of solid angle, $\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathrm{\Omega}=2\mathit{\pi}sin\mathrm{\chi}\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathrm{\chi}$, as

## C.1.3 Scattering for hard sphere, Coulomb, and gravitational potentials

In the case of hard spheres of diameter *D*, the impact parameter and the deflection angle are related by $b=Dsin{\mathit{\varphi}}_{0}=Dsin((\mathit{\pi}-\mathrm{\chi})/2)$. Replacing this expression in (C.6) gives

that is, after the collision, particles are scattered isotropically. This result is used in the statistical modelling of hard sphere collisions, where the postcollisional relative velocity is selected from an isotropic distribution.

Coulomb and self-gravitating systems are characterised by a potential $U=\mathit{\alpha}/r$, where the sign of $\mathit{\alpha}$ depends on the signs of the interacting charges for the Coulomb case and is always negative for the gravitational interaction. Using eqn (C.5), one gets, after some trigonometric transformations, ${\mathit{\varphi}}_{0}=arctan\left(\mathit{\mu}{g}^{2}b/\mathit{\alpha}\right)$. The deflection angle therefore satisfies

(p.238) which allows one to compute the differential cross section as

Notably, this does not depend on the sign of $\mathit{\alpha}$. This implies that, for Coulomb interactions, the statistical scattering properties do not depend on the signs of the charges.

For large separations, the deflection angle for gravitational interactions is $\mathrm{\chi}\approx 2\mathit{\alpha}/(\mathit{\mu}{g}^{2}b)$. After the collision, the relative velocity acquires a transversal component, $\mathit{\delta}{g}_{\perp}=gsin\mathrm{\chi}\approx -\frac{4Gm}{gb}$, where for particles of equal mass we have used that $\mathit{\alpha}=-Gmm$ and $\mathit{\mu}=m/2$. Finally, each of the colliding particles gets a transversal velocity component of

where we have used that ${\text{c}}_{2/1}=\text{V}\pm \text{g}/2$ and that the velocity of the centre of mass $\text{V}$ is unchanged. This result is used to derive the dynamical friction properties for self-gravitating systems in Chapter 6.

# C.2 Quantum mechanics

## C.2.1 Time-dependent perturbation theory

The method used in this book to study transitions or scattering processes between quantum states is time-dependent perturbation theory (Messiah, 2014, Cohen-Tannoudji *et al.*, 1992). Consider a system described by an unperturbed Hamiltonian ${H}_{0}$ whose eigenstates $|n\u27e9$ and energies ${\mathit{\epsilon}}_{n}$ are known,

Initially, the system is prepared in the pure state $|i\u27e9$, and at time $t=0$ a perturbation in the Hamiltonian is switched on, such that it now reads,

where we write $V(t)=\u03f5\stackrel{\u02c6}{V}(t)$ ($\u03f5\ll 1$) to indicate that the perturbation, which may depend on time, is small.

Our aim is to determine how the system evolves and the probability that, after some time, the system will be in a state $|k\u27e9$, different from the initial one. We assume that the perturbation is weak enough that the Hilbert space is unchanged and that the state of the system at any time $|\mathit{\psi}(t)\u27e9$ can be written in terms of the unperturbed states,

Inserting this expansion into the Schrödinger equation, $i\mathrm{\hslash}\frac{\phantom{\rule{thinmathspace}{0ex}}\text{d}}{\phantom{\rule{thinmathspace}{0ex}}\text{d}t}|\mathit{\psi}\u27e9=H|\mathit{\psi}\u27e9$, results in

(p.239) where we have used the orthogonality of the eigenstates, $\u27e8k|j\u27e9={\mathit{\delta}}_{jk}$.

We now look for perturbative solutions of the previous equation. We first note that, in the absence of the perturbation, the amplitudes are described simply by ${c}_{k}^{(0)}(t)={e}^{-i{\mathit{\epsilon}}_{k}t/\mathrm{\hslash}}{c}_{k}^{(0)}(0)$, which suggests that it will be convenient to perform the change of variables,

and it is expected that, for $\u03f5\ll 1$, the amplitudes *b _{k}* will evolve slowly in time.

^{[}

^{2}

^{]}In this case, we can expand

where now the time derivatives of *b _{k}* are of order 1. Replacing this expansion in (C.14) gives

As expected, the zeroth-order amplitudes state that the initial condition is preserved, ${b}_{k}^{(0)}={\mathit{\delta}}_{ik}$, where $|i\u27e9$ is the state in which the system was initially prepared. At first order, therefore, we get

from which the probability of finding the system in the state $|k\u27e9$, $|{b}_{k}{|}^{2}$, is readily obtained.

## C.2.2 Fermi golden rule

Two cases are of particular interest. The first consists of a constant perturbation $\stackrel{\u02c6}{V}$. In this case,

where we have replaced $\u03f5\stackrel{\u02c6}{V}=V$. The probability of finding the system in a state $|k\u27e9$ at a time *t* after the perturbation was switched on is then

When the final states are discrete, oscillations take place. However, the picture is completely different when they are continuous or quasicontinuous, as is the case for quantum gases studied in Chapters 7, 8, and 9. In order to continue working with normalised functions, we consider finite volumes where the quantum states are quasicontinuous. The
(p.240)
probabilities *p _{k}* are factored as a term that depends explicitly on the final state times the function $F(({\mathit{\epsilon}}_{k}-{\mathit{\epsilon}}_{i})/\mathrm{\hslash},t)$, which depends only on the energy of the final state. For quasicontinuous states we can label them as $|k\u27e9=|\mathit{\epsilon},\mathit{\mu}\u27e9$, where $\mathit{\mu}$ groups all the other quantum numbers for a given energy (for example, the direction of the momentum for free particles). Now, the probability of transit to a state in a window of $\mathit{\epsilon}$ and $\mathit{\mu}$ is

where *g* is the density of states and we have used the general rule, $\sum _{k}}={\displaystyle \sum _{\mathit{\mu}}}\int g(\mathit{\mu},\mathit{\epsilon})\phantom{\rule{thinmathspace}{0ex}}\text{d}\mathit{\epsilon$. For large times, the function *F* is peaked for $\mathit{\epsilon}$ close to ${\mathit{\epsilon}}_{i}$, and indeed it becomes a Dirac delta function, $\underset{t\to \mathrm{\infty}}{lim}}F((\mathit{\epsilon}-{\mathit{\epsilon}}_{i})/\mathrm{\hslash},t)=2\mathit{\pi}\mathrm{\hslash}t\mathit{\delta}(\mathit{\epsilon}-{\mathit{\epsilon}}_{i})$ (see Exercise C.3), which allows us to write,

The probability increases linearly with time, allowing us to define the probability transition rate from the initial state $|i\u27e9$ to a quasicontinuous state $|k\u27e9$ as

This important result, which states that, under constant perturbations, transitions can only take place between states with equal energy, is known as the Fermi golden rule.

The second case of interest is that of an oscillatory perturbation. As we are considering up to linear order in perturbation theory, it suffices to deal with monochromatic perturbations, $V(t)={V}_{\mathit{\omega}}sin(\mathit{\omega}t)$ or $V(t)={V}_{\mathit{\omega}}cos(\mathit{\omega}t)$. Both cases give, for long times, the transition rate,^{[}^{3}^{]}

This states that, under oscillatory perturbations, transitions can occur only between states that differ in energy by one quantum $\mathrm{\hslash}\mathit{\omega}$. The first term gives ${\mathit{\epsilon}}_{k}={\mathit{\epsilon}}_{i}+\mathrm{\hslash}\mathit{\omega}$, meaning that the system absorbs one quantum from the field, while for the second term, ${\mathit{\epsilon}}_{k}={\mathit{\epsilon}}_{i}-\mathrm{\hslash}\mathit{\omega}$, one quantum is emitted to the field. Note that both processes present equal transition rates.

# Exercises

(C.1)

**Collision rule for different masses**. Show that, in a binary collision of particles with different masses ${m}_{1}$ and ${m}_{2}$, the collision rule (C.2) is also satisfied. To show this, it is convenient to write the (p.241) energy and momentum conservation equations in terms of the relative and centre-of-mass velocities.(C.2)

**Oscillatory perturbation**. Derive eqn (C.25) for the perturbation $V(t)={V}_{\mathit{\omega}}sin(\mathit{\omega}t)$.(C.3)

**Dirac delta function**. Consider the function $f(x,t)=\frac{1}{\mathit{\pi}t}{\left[\frac{sin(xt)}{x}\right]}^{2}$. Show that, as a function of*x*, it is normalised to unity and that when $t\to \mathrm{\infty}$ it becomes concentrated around the origin, therefore becoming a Dirac delta distribution (see Fig. C.2).