|
|
[QUE/QFT-03004] QFT-PROBLEMNode id: 4012pageDefine Pauli Lubanski operator \(W_\sigma\) by \[ W_\sigma = -\frac{1}{2} \epsilon_{\mu\nu\lambda\sigma} M^{\mu\nu} P^\sigma\] where \(P^\mu\) is energy momentum four vector and \(M^{\mu\nu}\) angular momentum tensor. Prove the following relations
- \(W^\sigma P^\sigma =0\)
- \(\big[W^\sigma , P^\mu\big] =0 \)
- \(W_\sigma\) is a four vector, {\it i.e.} \(\big[M_{\mu\nu}, W_\sigma\big] = -i(W_\mu g_{\nu\sigma}-W_\nu g_{\mu\sigma})\)
- \( \big[W_\lambda, W_\sigma \big] = i\epsilon_{\lambda\sigma\alpha\beta}W^\alpha P^\beta\)
- \(P_\mu P^\mu\) and \(W^\sigma W_\sigma\) are Cashimir invariants of the Poincare group, they commute with all the ten generators of the Poincare group.
- Prove that \(W^2 = -\frac{1}{2} M_{\mu\nu} M^{\mu\nu} P^2 + M_{\mu\sigma}M^{\nu\sigma} P^\mu P_\sigma \)
|
|
22-02-06 18:02:04 |
n |
|
|
MM-Sub-01 Topics in Set TheoryNode id: 4579draftnode |
|
21-08-12 16:08:15 |
n |
|
|
[NOTES/QFT-04005] S Matrix in Interaction Picture qft-lsn-04005Node id: 3895page |
|
22-03-31 12:03:36 |
n |
|
|
Testing NODE 4387Node id: 5277page |
|
22-02-14 20:02:55 |
n |
|
|
[QUE/ME-12005] ME-PROBLEMNode id: 3975page
- A satellite of mass 2000 kg is to be put into a circular orbit around the earth of radius 1100 km. What is the minimum energy required?
- What will the minimum energy required to transfer it to an elliptic orbit having minimum and maximum distances 1100km and 4100km?
\(R_e=6400 \text{km},\quad GM/R_e^2=g\Rightarrow GM=gR_e^2\) Initial energy \(E_i\) of the satellite on the surface of the earth is \begin{eqnarray} E_i &=& -\frac{GMm}{R_e} = -\frac{gR_e^2m}{R_e}\\ &=& -gR_e^2m = (9.8 \text{(m/s}^2))(6400\times 1000 \text{\,m})(2000\text{\,kg})\\ &=& -9.8\times 128 \times10^8\\ &=&-12.5\times 10^{10} J \end{eqnarray} Final energy of the satellite in the orbit at height of 1100 km be \(E_f\). \begin{eqnarray} E_f &=& -\frac{GM,}{2(R+h)} = -\frac{gR_e^2 m}{2(R_e+h)}\\ &=& -\frac{9.8\times(6400\times 6400\times 10^6)\times 2000}{2\times7500\times10^3}\\ &=& \frac{9.8\times64\times 64}{2\times75}\times10^8\\ &=& -5.35\times10^{10} \end{eqnarray} Therefore the energy required to put the satellite in the circular orbit at a height 1100km \\ \(E_f-E_i=(-5.35 +!2.5 )\times 10^{10} \text{J}=7.15\times 10^{10} \text{J}\). \paragraph*{Elliptic orbit} Let \(r_1,r_2,a\) be, respectively, the perigee, apogee and the semi-major axis of the elliptic orbit. Then \(2a=r_1+ r_2\) It is give that \[r_1= 1100 km +R_e, r_2 =4100km+ R_e\] Therefore \[ 2a= 1100+4100 + 2\times 6400 = 1800\text{\,km}\] Hence a=9000km. The energy of the satellite in the elliptic orbit is \begin{eqnarray} E_\text{ell} &=& -\frac{GMm}{2a} = -\frac{gR_e^2m}{2a}\\ &=& -\frac{9.8 \times 6400\times6400\times10^6 \times 2000}{1800\times 10^3}\\ &=& -\frac{9.8\times64^2 \times 2\times 10^5}{18}\\ &=& -4.4 \times 10^{10} J. \end{eqnarray} Therefore energy requires to transfer the satellite from the circular orbit to the elliptic orbit is \((-4.4+5.35)\times 10^{10}\text{ J} = 9.3\times 10^{9}\text{J}.\)
|
|
22-02-08 18:02:38 |
n |
|
|
[QUE/QFT-14003] QFT-PROBLEMNode id: 4062page$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$
$\newcommand{\Lsc}{\mathscr{L}}$
For a self coupled scalar theory with interaction Lagrangian density given by \begin{equation} \Lsc_{\text{int}}= \frac{\lambda}{3!}:\phi(x)^3: \end{equation} Compute \begin{eqnarray} \int d^4y_1 \int d^4y_2\matrixelement{0}{T(\phi(x_1)\phi(x_2) \Lsc(y_1) \Lsc(y_2)}{0} \end{eqnarray}
|
|
22-02-01 19:02:34 |
n |
|
|
[LSN/Cm-02001] Euler Lagrange Equations of Motion Node id: 4357page |
|
22-03-31 00:03:07 |
n |
|
|
[QUE/CM-02023]Node id: 4425page |
|
22-03-19 17:03:05 |
n |
|
|
[QUE/ME-02019] ME-PROBLEMNode id: 3949pageShow that an rotation by an infinitesimal angle \(\Delta \theta\) about an axis \(hat{n}\) is equivalent to successive rotations by infinitesimal angles \(\alpha, \beta, \gamma\) about the three coordinate axes. Keeping first order terms in \(\Delta \theta\), find expressions for the angles \(\alpha, \beta, \gamma\) in terms of components of \(hat{n}\) and \(\Delta\theta\).
|
|
22-02-07 21:02:23 |
n |
|
|
[QUE/QFT-03006] QFT-PROBLEMNode id: 4014page
The matrix for an infinitesimal Lorentz boost along direction \(\hat{n}\) transformation can be written as \begin{equation} \Lambda = I + \Delta v (\hat{n}\cdot \vec{Y}), \end{equation} where the matrices \(\vec{Y}\) are given by \begin{equation} Y_1=\begin{pmatrix} 0 &1 & 0 & 0\\1 & 0 & 0& 0\\0 & 0&0&0\\0&0&0& 0\\ \end{pmatrix}\qquad; Y_2=\begin{pmatrix} 0 &0 & 1 & 0\\0 & 0 & 0& 0\\1& 0&0&0\\0&0&0& 0\\ \end{pmatrix}\qquad; Y_3=\begin{pmatrix} 0 &0 & 0 &1 \\0 & 0 & 0&0 \\0 & 0&0&0\\1&0&0& 0\\ \end{pmatrix}. \end{equation} Work out the transformation matrix \(\Lambda\) for a finite boost by velocity \(v\) by taking it as \(N\) successive transformations and considering the limit \(N\to \infty\). Hence show that \begin{eqnarray} x^{{'}\,0} &=& \cosh \alpha x^0 +\sinh\alpha (\hat{n}\cdot\vec{x})\\ \vec{x}\,^{'} &=& [\vec{x}- (\hat{n}\cdot \vec{x})\hat{n}] + \hat{n} [\cosh \alpha (\hat{n}\cdot\vec{x}) +\sinh \alpha x^0] \end{eqnarray} where \(\tanh\alpha=v\).
We first compute powers of \(\hat{n}\cdot\vec{Y}\), where \(\hat{n}=(n_1,n_2,n_3)\) is a unit vector.
\begin{eqnarray} \hat{n}\cdot\vec{Y} &=& \begin{pmatrix} 0 & n_1 & n_2 & n_3 \\ n_1 & 0 & 0 & 0\\ n_2 & 0 & 0 & 0\\n_3 & 0 & 0 & 0 \end{pmatrix}\\ (\hat{n}\cdot\vec{Y})^2 &=& \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & n_1^2 & n_1 n_2 & n_1 n_3 \\ 0 & n_2 n_1 & n_2^2 & n_2 n_3 \\ 0 & n_3n_1 & n_3 n_2 & n_3 n_1 \end{pmatrix} (\hat{n}\cdot\vec{Y})^3 &=& \hat{n}\cdot\vec{Y} \end{eqnarray} Hence we get \begin{eqnarray} \exp\big(-\omega \hat{n}\cdot \vec{Y} \big) &=& 1- \omega (\hat{n}\cdot\vec{Y}) + \frac{\omega^2}{2!}(\hat{n}\cdot\vec{Y})^2 - \frac{\omega^3}{3!} (\hat{n}\cdot\vec{Y})^3 + \frac{\omega^4}{4!}(\hat{n}\cdot\vec{Y}) -\frac{\omega^5}{5!}(\hat{n}\cdot\vec{Y}) + \ldots. \end{eqnarray} Using \eqRef{EQ07} we get \begin{eqnarray}\nonumber \exp(-\omega \hat{n}\cdot \vec{Y}) &=& I- (\hat{n}\cdot\vec{Y}) \Big[ (\omega +\frac{\omega^3}{3!}+ \frac{\omega^5}{5!} +\ldots\Big] + (\hat{n}\cdot\vec{Y})^2 \Big[ \frac{\omega^2}{2!} + \frac{\omega^4}{4!} + \frac{\omega^6}{6!}+ \ldots \Big]\\ &=& I - (\hat{n}\cdot\vec{Y})\sinh \omega + (\hat{n}\cdot\vec{Y})^2\, (\cosh \omega-1) \end{eqnarray} Using \EqRef{EQ05} and \eqRef{EQ06} we get \begin{equation} \exp(-\omega \hat{n}\cdot\vec{Y}) = \begin{pmatrix} \cosh \omega &-n_1 \sinh \omega & -n_2\sinh\omega &-n_3\sinh \omega\\ -n_1 \sinh\omega &1+n_1^2(\cosh\omega-1) &n_1n_2(\cosh \omega -1) &n_1n_3(\cosh \omega -1) \\ -n_2 \sinh\omega &n_2n_1(\cosh \omega -1) &1+n_2^2(\cosh\omega-1) & n_2n_3(\cosh \omega -1)\\ -n_3 \sinh \omega& n_3n_1(\cosh \omega -1)& n_3n_2(\cosh \omega -1)& 1+ n_3^2(\cosh\omega-1) \end{pmatrix} \end{equation} Writing \begin{equation} \begin{pmatrix} x^{0\,{'}} \\x^{1\,{'}}\\x^{2\,{'}}\\x^{3\, {'}} \end{pmatrix} = \Lambda \begin{pmatrix} x^0 \\x^1\\x^2\\x^3 \end{pmatrix} \end{equation} Performing the matrix multiplication in the right hand side we get \begin{eqnarray}\label{EQ12} x^{{'}\,0} &=& \cosh \omega - \sinh \omega (\hat{n}\cdot\vec{x})\\ \vec{x}\,{'} &=& -\sinh \omega\, x^0 + (\cosh \omega -1) (\hat{n}\cdot\vec{x}) \hat{n} + \vec{x}.\label{EQ13} \end{eqnarray} Taking dot product of both sides in \eqRef{EQ13}, we get
\begin{eqnarray} x^{{'}\,0} &=& \cosh \omega x^0 - \sinh \omega (\hat{n}\cdot\vec{x})\\ (\hat{n}\cdot\vec{x}\,{'}) &=& -\sinh \omega\, x^0 + \cosh \omega (\hat{n}\cdot\vec{x}) \end{eqnarray}
Identifying \(\tanh \omega = (v/c)\), the above form give the standard Lorentz boost along the direction \(\hat{n}\).
|
|
22-02-06 18:02:38 |
n |
|
|
[QUE/QFT-15009] QFT-PROBLEMNode id: 4399page$\newcommand{\Lsc}{\mathscr L}$
Assuming interactions of charged pions to be of the form \(\Lsc_\text{int} (x)= (g/4)(\pi(x)^+\pi(x)^-)^2\) find the \(S\) matrix element for \(\pi-\pi\) scattering \[\pi^+ + \pi^- \longrightarrow \pi^+ + \pi^-\] transition probability per unit time per unit volume for \(\pi - \pi\) scattering. Compute the total cross section for the scattering process and show that \[ \frac{d\sigma}{d\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]
|
|
22-02-01 19:02:32 |
n |
|
|
testing blank spaces Node id: 3902page |
|
20-12-15 10:12:02 |
n |
|
|
[QUE/ME-12008] ME-PROBLEMNode id: 3977pageConsider a particle of mass \(\mu\) moving in a potential \[ V(r) = \frac{1}{2}\mu\omega^2 r^2 +\frac{\lambda^2}{2\mu r^2}. \]
- Find condition on energy \(E\) and angular momentum \(L\) for circular orbits to exist.
- Does there exist a circular orbit for \(L=0\)?
- Assume orbital angular momentum \(L=0\), energy \(E= \frac{25}{2}\mu\omega^2a^2\), \(\lambda=12\mu\omega^2\) Use initial conditions \[ r(t)\big|_{t=0} = 4a;\quad \dot{r}(t)\big|_{t=0} = 0 \text{ and } \dot{\theta(t)}\big|_{t=0}=0 \] solve the equations of motion and obtain \(r\), \(\theta\) as function of time. Describe the motion that takes place under conditions specified here.
|
|
22-02-08 18:02:47 |
n |
|
|
[QUE/QFT-15002] QFT-PROBLEMNode id: 4065page$\newcommand{\Hsc}{\mathscr H}$
The interaction of \(\Lambda^0\) hyperon, responsible for decay into a proton and a \(\pi^-\), is given by \[ \Hsc_\text{int} = \bar{\psi}_p(x) ( g - g^\prime\gamma_5)\psi_\Lambda(x) \phi_\pi ^\dagger + h.c. \]
- Give examples of three virtual processes allowed in the first order of this interaction term.}
- Show that the partial decay rate of \(\Lambda^0 \longrightarrow p + \pi^-\) is given by \[ \Gamma = \frac{1}{4\pi}\frac{|\vec{p}|}{M_\Lambda}\left(|g|^2(E_p+M_p) + |g^\prime|^2 (E_p-M_p) \right)\]
|
|
22-02-06 19:02:29 |
n |
|
|
[QUE/CM-02025]Node id: 4427page |
|
22-03-19 18:03:36 |
n |
|
|
[QUE/ME-02021] ME-PROBLEMNode id: 3951page
A rotation takes a vector by an angle \(\alpha\) about axis (2,1,2) takes vector \(\underline{\sf A}\) to new vector \(\underline{\sf A}{'}\). Taking \(\alpha =\cos^{-1}\Big(\frac{3}{5}\Big), 0 < \alpha < \pi/2 \), find the rotation matrix \(R\), such that \(\underline{\sf B}=R \underline{\sf A}\) that relates two vectors.
|
|
22-02-07 21:02:17 |
n |
|
|
[QUE/QFT-04002] QFT-PROBLEMNode id: 4016pageCompute unequal time commutator \[ \big[\psi(x,t), \psi(y,t{'})\big]\] where the Schrodinger field \(\psi(x,t)\) obey free particle Schrodinger equation.
|
|
22-02-04 21:02:52 |
n |
|
|
[QUE/QFT-15012] QFT-PROBLEMNode id: 4401page$\newcommand{\Lsc}{\mathscr L}$
Consider a system of two real scalar fields \(\phi_1, \phi_2\) described by the Lagrangian density \begin{equation} \Lsc = \frac{1}{2} \partial_\mu \phi_i \partial^\mu \phi_i - \frac{1}{2} m^2 \phi_i\phi_i -\frac{1}{4} \lambda (\phi_i\phi_i)^2. \end{equation} Compute the scattering cross section to the lowest order in \(\lambda\). Find the cross sections for the three processes
- \(\phi_1 + \phi_2 \longrightarrow \phi_1 +\phi_2\)
- \(\phi_1 + \phi_1 \longrightarrow \phi_1 +\phi_1\)
- \(\phi_1 + \phi_1 \longrightarrow \phi_2 +\phi_2\)
Write your answers as a constant times \(\sigma_0\equiv \frac{\lambda^2}{64\pi s}\) where \(s\) is the total energy in the center of mass frame
|
|
22-01-31 08:01:50 |
n |
|
|
[QUE/ME-12010] ME-PROBLEMNode id: 3979pageShow that the radius of circular orbit for energy \(E\) and angular momentum \(L\) is given by \(R=\frac{L^2}{k\mu}\).
|
|
22-02-08 18:02:42 |
n |
|
|
[QUE/QFT-15004] QFT-PROBLEMNode id: 4067pageThe original four fermion interaction for beta decay of neutron \[n \longrightarrow p + e^- + \bar{\nu} \] is of the of form \[ \bar{\psi}_p(x)\gamma_\mu\psi_n(x) \bar{\psi}_\nu(x) \gamma^\mu \psi_e(x) + h.c.\] Now consider other processes given below. Which of these processes (real or virtual) are permitted and which ones are not permitted by the above interaction in the first order?
- \( \bar{p} \longrightarrow \bar{n} + e^- +\bar{\nu} \);
- \( \bar{p} \longrightarrow \bar{n} + e^- +\nu \);
- \( n \longrightarrow p + e^+ + \nu \);
- \( p \longrightarrow n + e^+ + \bar{\nu} \);
- \( \bar{n} \longrightarrow \bar{p} + e^+ + \nu \);
- \( \bar{n} \longrightarrow \bar{p} + e^+ + \bar{\nu} \).
Give brief reason in each case.
|
|
22-02-06 19:02:43 |
n |
||Message]