## E. G. Peter Rowe

## Chapter 3

## Asymptotic Momentum Conservation - all with Video Answers

## Educators

Chapter Questions

How fast, with respect to a frame $K$, must a particle move so its kinetic energy (with respect to $K$ ) equals its rest energy?

Guilherme Barros

Numerade Educator

A particle of mass $m$ decays from rest (in $K$ ) into a particle of mass $m^{\prime}$ and a photon $\gamma$. Show that the $K$-energies of the end products are $E^{\prime}=\left(m^2+m^{\prime 2}\right) / 2 m$, and $E_\gamma=\left(m^2-m^{\prime 2}\right) / 2 m$.

Daniel Sneed

Numerade Educator

If $k$ is a lightlike momentum vector, verify that for every positive number $E$ there is at least one inertial frame $K$ such that the associated $K$-energy is $E$.

Narayan Hari

Numerade Educator

A particle $A$, of unit mass, decays from rest in $K$ into two particles $B$ and $C$. The speeds $V_B$ and $V_C$ of $B$ and $C$ with respect to $K$ are measured and have $\gamma$-factors $\gamma_B=2, \gamma_C=3\left(\gamma=\left(1-V^2\right)^{-1 / 2}\right)$. What are the masses of $B$ and $C$ ?

Mayank Tripathi

Numerade Educator

We use the usual boost-related frames $K$ and $K^{\prime}$. In $K$ there is a uniform spatial distribution of particles of mass $m \neq 0$, all moving, at speed $U$ with respect to $K$, in the direction $\cos \theta \boldsymbol{E}_1+\sin \theta \boldsymbol{E}_2$. Show that the relation between the $K^{\prime}$-number density $n^{\prime}$ and the $K$-number density $n$ is

$$

n^{\prime}=n \gamma(1-U V \cos \theta) .

$$

(Hint: since the particles are massive one can relate each density to the rest frame density.)

Narayan Hari

Numerade Educator

We use the usual boost-related frames $K$ and $K^{\prime}$. In $K$ there is a uniform spatial distribution of photons, all moving in the $\cos \theta \boldsymbol{E}_1+\sin \theta \boldsymbol{E}_2$ direction. Show that the relation between the $K$-number density $n$ (at $t=0$, say) and the $K^{\prime}$-number density $n^{\prime}$ (at $t^{\prime}=0$ ) is

$$

n^{\prime}=n \gamma(1-V \cos \theta),

$$

where $V$ is the speed of $K^{\prime}$ with respect to $K$. (This result was worked out by several people in 1968 to use in looking for a directional effect in the background cosmic radiation.)

Eduard Sanchez

Numerade Educator

An atomic transition occurs with the emission of a photon, $A \rightarrow B+\gamma$, the rest masses being $M_A$ and $M_B$.

a) If $A$ is initially at rest in the lab, find the lab energy $E$ of the photon and the speed $V$ of recoil (with respect to the lab) of $B$.

b) What lab energy $E^{\prime}$ would a (different) photon have to have to stimulate the reverse transition $\gamma+B \rightarrow A$ if the state $B$ is initially at rest in the lab? If the photon's energy $E^{\prime \prime}<E^{\prime}$, show that the minimum speed $V^{\prime \prime}$ (with respect to the lab) with which $B$ would have to move for the transition to occur is

$$

V^{\prime \prime}=\frac{\left(M_A^2-M_B^2\right)^2-4\left(E^{\prime \prime} M_B\right)^2}{\left(M_A^2-M_B^2\right)^2+4\left(E^{\prime \prime} M_B\right)^2} .

$$

Mayukh Banik

Numerade Educator

If a particle of mass $m$ collides with a particle of mass $M$ to form a particle of mass $M+m$, the relative speed of the collision must be zero. (In general the mass of the compound particle determines the relative speed, and $M+m$ is the threshold value of the compound mass.)

Farhanul Hasan

Numerade Educator

A particle $A$ of mass $M$ decays from rest (with respect to frame $K$ ) into a particle $B$ of mass $m$ and a particle $C$ of mass $\mu$.

a) Find expressions in terms of the masses alone for the $K$-energies of $B$ and $C$.

b) Show that the two decay products $B$ and $C$ are separated by a $K$ distance

$$

\frac{2 M^2 T \sqrt{\lambda\left(M^2, m^2, \mu^2\right)}}{M^4-\left(m^2-\mu^2\right)^2}

$$

at a $K$-time interval $T$ after the decay event.

c) Find the $\bar{K}_C$-distance between $B$ and $C$ at a $\bar{K}_C$-time $T_C$ after the decay event, where $\tilde{K}_C$ is the rest frame of $C$.

Keshav Singh

Numerade Educator

Frames $K$ and $K^{\prime}$ are in standard boost relation. If light of frequency $\nu$ in $K$ travels in a direction inclined at an angle $\psi$ to the $+x$-axis, what is its frequency and direction with respect to $K^{\prime}$ ? (This is the Doppler effect for the case when the relative direction of the frames is different from the direction of the light.)

Khoobchandra Agrawal

Numerade Educator

In a Compton scattering the ingoing photon has frequency $\nu$ and the scattered photon $\nu^{\prime}$, both with respect to the lab, where the target electron is at rest. If the electron recoils with lab-speed $V$ at a lab-angle $\psi$ to the direction of the incoming photon, show that

$$

\frac{\nu^{\prime}}{\nu}=\frac{1-V \cos \psi}{\sqrt{1-V^2}} \text {. }

$$

(Note that in this question the photons with frequencies $\nu$ and $\nu^{\prime}$ are different, whereas in the previous question there is only one sort of photon viewed from different frames $K$ and $K^{\prime}$.)

Suzanne W.

Numerade Educator

A particle $A$ of mass $m_A$ is struck by an incident photon; the result of the collision is a particle $B$ of mass $m_B$ and a scattered photon. Show that if the frequency of the incident photon in the rest frame of $A$ is $f$, then the frequency of the scattered photon in the rest frame of $B$ is $\left(m_A / m_B\right) f$.

Eduard Sanchez

Numerade Educator

In a Compton experiment the initial photon has energy $E_\gamma$ in the lab, where the target electron, of mass $m$, is initially at rest. Show that the CM-frame moves at speed $V=E_\gamma /\left(E_\gamma+m\right)$ with respect to the lab and find the CM-energies for the initial photon and initial electron. What is the CM-energy of the photon after scattering? If the scattering angle of the photon in the lab is $\theta$, what is it in the CM-frame?

Suzanne W.

Numerade Educator

A disc rotates with angular velocity $\omega$ with respect to the laboratory. On its rim, at radius $R$, is an emitter $E$, and at angular separation $\alpha$ there is an absorber $A$. The quantities $\omega, \alpha, R$ are all derived from lab measurements. A photon is emitted from $E$ at just the right angle to be absorbed by $A$ when the photon meets the rim again. Show that the frequency of the photon in the rest frame of $E$ at the moment of emission is equal to its frequency in the rest frame of $A$ at the moment of absorption.

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Two particles, of masses $m_1$ and $m_2$ and four-momenta $p_1$ and $p_2$, approach each other. Define a centre-of-momentum (CM) frame for the particles. Obtain expressions for the energies $E_1$ and $E_2$ and relative velocities $\boldsymbol{V}_1$ and $\boldsymbol{V}_2$ of each particle with respect to the CM-frame. Verify that

$$

\boldsymbol{V}_1=-\boldsymbol{V}_2\left(m_2^2-\boldsymbol{p}_1 \cdot \boldsymbol{p}_2\right) /\left(m_1^2-\boldsymbol{p}_1 \cdot \boldsymbol{p}_2\right) .

$$

If the particles interact and emerge with the same masses but momenta $\boldsymbol{p}_1^{\prime}, \boldsymbol{p}_2^{\prime}$, show that the new CM-energies are the same as before, and the magnitudes of the new relative velocities are the same too.

Keshav Singh

Numerade Educator

The worldlines $L_1$ and $L_2$ of two nonintersecting, freely moving massive particles are parallel to their momentum vectors $\boldsymbol{p}_1=m_1 \boldsymbol{v}_1$ and $\boldsymbol{p}_2=$ $m_2 v_2$. Let $O_1$ and $O_2$ be arbitrary origins on $L_1$ and $L_2$, respectively. Other points on each worldline can then be parameterised with their proper times, $\overrightarrow{O_1 P_1}=\tau_1 v_1, \overrightarrow{O_2 P_2}=\tau_2 v_2$. Show that the point $S_2$ on $L_2$ which is simultaneous in the CM frame with $O_1$ on $L_1$ has proper time $\tau_2=-m_2\left(\boldsymbol{p} \cdot \overrightarrow{O_1 O_2}\right) /\left(\boldsymbol{p} \cdot \boldsymbol{p}_2\right)$, where $\boldsymbol{p}$ is the total momentum. Show that the minimum distance in the CM frame, $\left|\overrightarrow{Q_1 Q_2}\right|$, between the two worldlines, is given by

$$

\overrightarrow{Q_1 Q_2} \cdot \overrightarrow{Q_1 Q_2}=\overrightarrow{O_1 S_2} \cdot \overrightarrow{O_1 S_2}-\left(\overrightarrow{O_1 S_2} \cdot \boldsymbol{q}\right)^2 /(\boldsymbol{q} \cdot \boldsymbol{q}) \text {, }

$$

where $\boldsymbol{q} \equiv\left(\boldsymbol{p}_1 \wedge \boldsymbol{p}_2\right) \cdot \boldsymbol{p} \equiv \boldsymbol{p}_1\left(\boldsymbol{p}_2 \cdot \boldsymbol{p}\right)-\boldsymbol{p}_2\left(\boldsymbol{p}_1 \cdot \boldsymbol{p}\right)$.

Khoobchandra Agrawal

Numerade Educator

An electron-positron pair annihilates into two photons. Suppose that in the CM-frame $K$ the direction of motion of the photons is perpendicular to the direction of motion of the electron. Show that in a frame $K^{\prime}$ moving at speed $V$ with respect to $K$ in the direction of the electron's motion the angle between the directions of the two photons is $\theta$ given by $\cos \theta=2 V^2-1$.

Penny Riley

Numerade Educator

As the headlight effect exemplifies, the angle between the spatial directions of two photons depends on the frame of reference. Two photons which are moving parallel with respect to one frame, however, move parallel with respect to any other frame. This property is not true of massive particles. Explain and give an example.

Manish Jain

Numerade Educator

In the reaction $\pi^{-}+p^{+} \longrightarrow K^{\circ}+A^{\circ}$ the target proton is at rest in the lab frame.

a) Using the masses $m_{\pi^{-}}=0.140 \mathrm{Gev}, m_p=0.938 \mathrm{Gev}, m_{K^{\circ}}=$ $0.498 \mathrm{Gev}, m_{A^{\circ}}=1.116 \mathrm{Gev}$ show that the total lab energy of the pion at threshold is 0.909 Gev.

b) In an experiment in which the lab three-momentum of the pions has magnitude 2.50 Gev the $\Lambda^{\circ}$ are observed to have three-momentum 0.60 Gev at a lab angle $45^{\circ}$ with respect to the incident pions. What is the velocity of the centre-of-momentum frame in this case? What is the magnitude of the three-momentum of the $K^{\circ}$ in the lab frame, and in the CM-frame?

Suzanne W.

Numerade Educator

Two particles $A$ and $B$ approach each other, one with mass $m_A$ and spacetime velocity $v_A$, the other with mass $m_B$ and spacetime velocity $v_B$. Show that the relative velocity of the centre-of-momentum frame with respect to the rest frame of $B$ may be written

$$

\frac{m_A v_A+m_A\left(\boldsymbol{v}_A \cdot \boldsymbol{v}_B\right) v_B}{m_B-m_A\left(\boldsymbol{v}_A \cdot \boldsymbol{v}_B\right)} \text {. }

$$

Khoobchandra Agrawal

Numerade Educator

A photon strikes a particle, of mass $m$, at rest in $K$. The products of the collision are three particles, each of mass $m$. Show that the $K$-energy of the photon is at least 4 m .

Arpit Gupta

Numerade Educator

A particle of mass $M$, at rest in $K$, decays into a particle of mass $m$, a particle of mass $\mu$ and a photon. Show that the maximum energy (with respect to $K$ ) that the photon can have is $\left[M^2-(m+\mu)^2\right] /(2 M)$.

Daniel Sneed

Numerade Educator

A particle of mass $m$ collides elastically with a stationary (with respect to $K$ ) particle of equal mass. ("Elastic" means that the end products of the process have the same masses as the original particles.) The incident particle has $K$-kinetic energy $T$ and it is scattered in the collision by a $K$-angle $\theta$. Show that its $K$-kinetic energy after the collision is

$$

T^{\prime}=2 m T \cos ^2 \theta /\left(2 m+T \sin ^2 \theta\right) .

$$

Naman Kumar

Numerade Educator

Two particles, each of mass $m$, collide elastically. If one of the particles is at rest in $K$ before collision and the other has total $K$-energy $E$, and after the collision the two particles move (with respect to $K$ ) with the same speed, show that the angle between their directions of motion is $\chi_1$ where

$$

\cos ^2(\chi / 2)=\left(E^2-m^2\right) /\left[(E+m)^2-4 m^2\right]=(E+m) /(E+3 m) .

$$

In the nonrelativistic limit, $\chi \rightarrow \pi / 2$.

Penny Riley

Numerade Educator

Two particles, of masses $\mu$ and $m$, scatter elastically. Momentum conservation is expressed by $\boldsymbol{k}+\boldsymbol{p}=\boldsymbol{k}^{\prime}+\boldsymbol{p}^{\prime}$. The Breit frame (or brick wall BW-frame) is defined by having its timelike basis vector $\boldsymbol{E}_0$ in the direction of $\boldsymbol{k}+\boldsymbol{k}^{\prime}$. The energies and spatial momenta with respect to this frame are $\epsilon, \boldsymbol{K} ; \epsilon^{\prime}, \boldsymbol{K}^{\prime} ; E, \boldsymbol{P} ; E^{\prime}, \boldsymbol{P}^{\prime}$. Show that $\epsilon=\epsilon^{\prime}, \boldsymbol{K}+\boldsymbol{K}^{\prime}=$ $\mathbf{0}, E=E^{\prime}, 2 \boldsymbol{K}=\boldsymbol{P}^{\prime}-\boldsymbol{P}$. If $t$ is defined by $t=-\left(\boldsymbol{k}-\boldsymbol{k}^{\prime}\right)^2$, show that $t=-4 \boldsymbol{K} \cdot \boldsymbol{K}$, and that the angle $\theta_B$ between $\boldsymbol{P}$ and $\boldsymbol{P}^{\prime}$ is given by $\cos \theta_B=1+\frac{1}{2} t /\left(E^2-m^2\right)$.

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Two particles of equal rest mass scatter elastically (same particle types after the collision as before). Before the collision the projectile has speed $V$ with respect to lab and $\gamma$-factor $\gamma$, while the target particle is at rest. After the collision the projectile goes off at an angle $\theta_1$ with respect to its incident direction and its new velocity has $\gamma$-factor $\gamma_1$. The target particle goes off at an angle $\theta_2$ with $\gamma$-factor $\gamma_2$. Show that for $i=1,2$

$$

\cos ^2 \theta_i=\frac{\gamma_i-1}{\gamma_i+1} \frac{\gamma+1}{\gamma-1}

$$

and hence that $\tan \theta_1 \tan \theta_2=\frac{2}{\gamma+1}$.

Kai Chen

Princeton University

Suppose a particle of mass $M$ decays into a particle of mass $m$ and a particle of mass $\mu$. If $Q$ is the three-momentum of the $m$-particle in the rest frame of the $M$-particle, and $K$ is the three-momentum of the $\mu$-particle in the rest frame of the $m$-particle, show that $m^2 \boldsymbol{K} \cdot \boldsymbol{K}=$ $M^2 Q \cdot Q$.

Daniel Sneed

Numerade Educator

Two particles, one of mass $m_1$ and momentum $\boldsymbol{p}_1$, the other $m_2, \boldsymbol{p}_2$, travel freely. Denote the total four-momentum by $\boldsymbol{K}$ and $\boldsymbol{K} \cdot \boldsymbol{K}=-M^2$. Show that the unit spatial vector in the CM-frame which points in the direction of the relative three-momentum (with respect to CM) of the first particle is

$$

\boldsymbol{q}=\frac{M}{\sqrt{\lambda}}\left[\boldsymbol{p}_1-\boldsymbol{p}_2-\frac{\left(m_1^2-m_2^2\right)}{M^2} \boldsymbol{K}\right],

$$

where $\lambda$ is the standard function $\lambda\left(M^2, m_1^2, m_2^2\right) \equiv M^4+m_1^4+m_2^4-$ $2 m_1^2 m_2^2-2 M^2\left(m_1^2+m_2^2\right)$. Write $\boldsymbol{p}_1$ and $\boldsymbol{p}_2$ in terms of $\boldsymbol{K}$ and $\boldsymbol{q}$.

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If a particle of mass $m$ collides elastically with a particle of mass $M$, the relative speed of one particle with respect to the other is the same after the collision as before.

Joseph Petrullo

Numerade Educator

Two particles of equal mass collide head-on (the collision takes place in one straight spatial line in $K$, the rest frame of one of the particles before the collision). Show that if $V$ is the initial relative speed of the projectile with respect to $K$, then after the collision the target recoils with speed $V$ (with respect to $K$ ) and the projectile is reduced to rest.

Chai Santi

Numerade Educator

A projectile particle, of mass $m$, collides head-on with a target particle of mass $M$, which is initially at rest in $K$. If $V$ is the initial speed of the projectile with respect to $K$, show that $U$, the recoil speed of the target (with respect to $K$ ), is given by

$$

\frac{1}{\sqrt{1-U^2}}=\frac{(\gamma+x)^2+\gamma^2-1}{(\gamma+x)^2-\gamma^2+1}

$$

where $x=M / m$ and $\gamma^{-2}=1-V^2$. (Hint: bear in mind the results of the previous two problems, the second of which is the case of equal masses.)

Naresh Bagrecha

Numerade Educator

In a high energy collider an electron and positron, of equal Lab energies, meet head on and produce a pair of heavy particles, $W^{+}$and $W^{-}$, of equal mass $M$. In this experiment the Lab is also the CM-frame, and the total energy is determined by $s \equiv-\left(\boldsymbol{p}_{+}+\boldsymbol{p}_{-}\right) \cdot\left(\boldsymbol{p}_{+}+\boldsymbol{p}_{-}\right)$. Suppose the $W^{+}$decays to $a+b$ and the $W^{-}$decays to $c+d$. Show that, in the approximation in which all particles $a, b, c, d$ are massless, the Lab energy $E$ of any one of them is bounded: $E_{-} \leq E \leq E_{+}$, where

$$

E_{ \pm}=\frac{1}{4}\left(\sqrt{s} \pm \sqrt{s-4 M^2}\right) .

$$

This bound can be used to measure the mass $M$ of the $W^{ \pm}$particles. (Note that if the $W^{ \pm}$are not assumed to mediate the process $e^{+} e^{-} \rightarrow$ abcd, the bound on the energy of any of the decay products would be simply $0 \leq E \leq \sqrt{s} / 2$.)

Narayan Hari

Numerade Educator