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EPP-Problem --- Id:: QUE/EPP-01003

Node id: 2388page

Consider a single particle production process \[ A + B \longrightarrow A + B + C.\] How many independent Lorentz invariant (Lorentz scalars) kinemtaic variables can be written down in terms of the four momenta of the particles, after taking into account of energy momentum and mass constraints.

Generalize the above result when there are \(n\) particles in the final state. Does your number give all possible Lorentz invariants?

kapoor

21-12-23 19:12:00

EPP-Problem --- Id:: QUE/EPP-01005

Node id: 2630page

The charged pion. \(\pi^+\) decays into a muon and a neutrino. \[ \pi^+ \longrightarrow \mu^+ + \nu.\] In the rest frame the muon momentum is \(|\vec{p}|=29.80\) MeV/c and mass of the muon is \(m_\mu=105.653\pm 0.002\) eV. Determine the mass of the charged pion.

kapoor

21-12-23 19:12:55

EPP-Problem --- Id:: QUE/EPP-01004

Node id: 2629page

The decay of a massive particle, mass \(M\), into two particles \(B\) and \(C\), masses \(m_1,m_2\), is not possible when \(M < m_1+m_2\). Prove this by choosing an appropriate Lorentz frame and applying energy momentum conservation. Is your argument valid when mass of the decaying particle is zero, \(M=0\)?
Show that a zero mass particle, such as a photon, cannot decay into two or more massive particles. Thus showing that a process such as \[ \gamma \longrightarrow e^+ \quad + \quad e^-\] is not possible for free photons in vacuum.

kapoor

21-12-23 19:12:03

EPP-Problem --- Id:: QUE/EPP-01002

Node id: 2387page

A particle of mass \(M\) decays into three particles of masses \(m_1,m_2,m_3\): \[A \longrightarrow B + C + D \] Considering the decay process in the rest frame of the particle \(A\), determine the angle between the momenta of \(B\) and \(C\) and hence show that the energies \(E_1\) and \(E_2\) must lie in a region of \(E_1,E_2\) plane bounded by the curve

\[ 4(E_1^2-m_1^2)(E_2^2-m_2^2) = (M^2-2m_1(E_1+E_2) +2E_1E_2 - m_3^2 + m_1^2 + m_2^2)^2\]

kapoor

21-12-23 19:12:57

EPP-Problem --- Id QUE/EPP-01001

Node id: 2386page

Consider the two body decay in an arbitrary frame

A	\(\longrightarrow\)	B	+	C
\(M\)		\(m_1\)		\(m_2\)
\((E,\vec{P})\)		\((\omega_1, \vec{k}_1)\)		\((\omega_2, \vec{k}_2).\)

Show that the angle \(\theta\) between the decay products is given by\begin{equation}\label{EQ01} \cos \theta =\frac{2\omega_1\omega_2-2m_1m_2}{2k_1k_2} + \frac{(m_1+m_2)^2-M^2}{2k_1k_2}\end{equation}
Use this result to prove that for pion decay \( \pi^0 \longrightarrow 2 \gamma\) in flight having velocity \(v\), the angle between the two photons is given by \[\cos (\theta/2) = v/2 .\]
Derive the condition \(M \ge m_1+m_2\) for a massive particle of mass \(M\) to decay in two particles.
Use the above result in \eqref{EQ01} to show that a massless particle cannot decay into two massive particles, even though the energy considerations appear to allow the decay.

kapoor

21-12-23 19:12:01

2021 Thermodynamics@CMI -- by H. S. Mani --- Links to VIdeo Recordings

Node id: 4901video_page

kapoor

21-12-20 08:12:59

Repository :: Mixed-Lot of Problems ---- [ALL-AREAS ]

Node id: 5001collection

WORK IN PROGRESS

kapoor

21-12-15 21:12:34

Quantum Field Theory --- Notes for Lectures and Problems --- MIXED-LOT

Node id: 5011multi_level_page

WORK IN PROGRESS

AK-47

21-12-13 19:12:09

Statistical Mechanics-Courses

Node id: 4953multi_level_page

WORK IN PROGRESS

kapoor

21-12-12 23:12:29

Quantum Information and Quantum Computation [QIQC-Home]

Node id: 5012multi_level_page

WORKING ON THIS PAGE IN PROGRESS

kapoor

21-12-12 17:12:09

Thermodynamics-Courses and Lectures

Node id: 4944multi_level_page

kapoor

21-12-12 16:12:28

Equation formatting and numbering

Node id: 383page

LaTeX has useful environments for multi-line equations, equation grouping, and alignment. Here are a few examples involving split, multiline, gather and align.

[To see the LaTeX code of any equation, click 'Show TeX' button above.]

ranjan

21-12-08 23:12:43