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Title+Summary	Name	Date
[QUE/EPP-01019] Node id: 2644page Consider electron proton scattering at a given kinetic energy \(T_L\) of the electron in the laboratory frame. Find the accessible energy range of \(t\), the four momentum transfer squared. Repeat the problem for the case of a proton beam with an electron target. Use the uncertainty relation to estimate the distance probed by the electron.Estimate what energy electrons would be required to probe down to 0.1 F	kapoor	22-04-18 19:04:07	n
[QUE/EPP-01013] Node id: 2638page Show that the sign of time component of a time like vector remains the same in all Lorentz frames. Is this statement true for space like vectors?	kapoor	22-04-18 19:04:57	n
[QUE/EPP-01014] Node id: 2639page If \(S^\mu\) is a space like vector show that there exists a frame in which the time component \(S^0\) vanishes. What is the corresponding statement for a time like vector?	kapoor	22-04-18 19:04:56	n
[QUE/EPP-01015] Node id: 2640page If \(A^\mu\) is a time like four vector show that there exist three independent vectors \(B^\mu\) such that \(B^\mu A_\mu=0\) and prove that all the three vectors are necessarily space like. Extend the above result to the case when \(A^\mu\) is (i) space like; (ii) light like.	kapoor	22-04-18 19:04:12	n
[QUE/EPP-01016] Node id: 2641page The polarization vectors of photon with four momentum \(k\), \(\vec{\varepsilon}(\vec{k},\lambda), \lambda=1,2\), satisfy \[ \vec{k}\cdot \vec{\varepsilon}(k, \lambda) = 0 \] Taking \(\eta_\mu=(1,0,0,0),\) and \[\varepsilon_{\mu}(\vec{k},\lambda)= (0,\vec{\varepsilon(k,\lambda)}), \quad \lambda=1,2, \] find a four vector \(B^\mu\) such that it is orthogonal to \(\varepsilon(k, \lambda)\) and \(\eta_\mu \). Hint: What should the required vector \(B_\mu\) be? Light like, time like or space like? Use the above result to show that \begin{eqnarray} \sum_{\lambda=1}^2 \varepsilon_\mu(k,\lambda) \varepsilon_\nu(k,\lambda) &=& - g_{\mu\nu} + \eta_\mu\eta_\nu - \frac{k_\mu k_\nu}{(k\cdot\eta)^2-k^2} \nonumber\\ &\qquad &+ \frac{(k\cdot\eta)(\eta_\mu k_\nu+ k_\mu\eta_\nu)}{(k\cdot\eta)^2-k^2} - \frac{(k\cdot\eta)^2\eta_\mu\eta_\nu}{(k\cdot\eta)^2-k^2}.\nonumber \end{eqnarray}	kapoor	22-04-18 19:04:17	n
[QUE/EPP-01017] Node id: 2642page If \(A^\mu, B^\mu\) are time like, what can you say about the four vector \( A^\mu + B^\mu \)? Will it always be time like or not?	kapoor	22-04-18 19:04:28	n
[QUE/EPP-01018] Node id: 2643page \(Z^0\) particles are to be produced in \(\bar{p} p\) collisions. What is the minimum energy required for (a) proton and for the anti-protons in c.m. frame , (b) for anti proton in lab frame ( proton at rest)?	kapoor	22-04-18 19:04:31	n
[QUE/EPP-01020] Node id: 2645page For each of the following decays, compute the momenta of the decay products in the rest frame of the decaying particle. \( Z^0 \longrightarrow e^+\quad + \quad e^-\) \( W^+ \longrightarrow e^+\quad + \quad \nu\) \( \Sigma \longrightarrow N \quad+ \quad\pi \) Google and search internet to know the masses of various particles in the three decays.	kapoor	22-04-18 19:04:48	n
[QUE/EPP-01025] Node id: 2662page If \(A_{\lambda\sigma}\) and \(B_{\lambda\sigma}\) are two second rank tensors satisfying \[ A_{\lambda\sigma}x^\lambda x^\sigma = B_{\lambda\sigma}x^\lambda x^\sigma\] for arbitrary four vector \(x^\mu\), what is the relationship between \(A_{\lambda\sigma}\) and \( B^{\lambda\sigma}\)?	kapoor	22-04-18 19:04:15	n
[QUE/EPP-01028] Node id: 2666page If \(A^\mu\) and \(B^\mu\) are light like four vector show that \(B^\mu A_\mu=0\) if and only if \[ A^\mu = \text{constant} \times B^\mu\]	kapoor	22-04-18 19:04:11	n
[QUE/EPP-01027] Node id: 2665page If \(A^\mu\) is a light like four vector show that there exist a light like and two time like vectors \(B^\mu\) such that \(B^\mu A_\mu=0\). Extend the above result to the case when \(A^\mu\) is (a) space like; (b) light like.	kapoor	22-04-18 19:04:58	n
[QUE/EPP-01027] Node id: 2664page If \(A^\mu\) is a light like four vector show that there exist a light like and two time like vectors \(B^\mu\) such that \(B^\mu A_\mu=0\). Extend the above result to the case when \(A^\mu\) is (a) space like; (b) light like.	kapoor	22-04-18 19:04:44	n
[QUE/EPP-01026] Node id: 2663page Let \(\epsilon\) and \(\lambda\) be two light like fur vectors. What is the range of allowed values of \(\epsilon\cdot\lambda\)? Express your answer in terms of \(\epsilon^0\) and \(\lambda^0\).	kapoor	22-04-18 19:04:52	n
[QUE/EPP-01024] Node id: 2661page Consider \(S_{\mu\nu}\) to be a second rank tensor having the following properties: \(S_{\mu\nu}\) is symmetric : \( S_{\mu\nu} = S_{\nu\mu} \); \(S_{\mu\nu}\) is antisymmetric : \( S_{\mu\nu} = - S_{\nu\mu} \); \(S\) is traceless:\( g^{\mu\nu} S_{\mu\nu} = 0\). Show that the above properties are remain invariant under Lorentz transformations.	kapoor	22-04-18 19:04:41	n
[QUE/EPP-01023] Node id: 2660page If \(p^\mu=(p^0,0,0,p)\) with \( p^{02} -\vec{p}^2 = m^2\), find all vectors \(\epsilon^\mu\) such that \(\epsilon^\mu p_\mu=0\). How many linearly independent vectors can you make?	kapoor	22-04-18 19:04:50	n
[QUE/EPP-01022] Node id: 2659page Find the charge \(Q\) which will exert the same force on an electron as Sun's gravitational pull on the earth when the distance between the charge \(Q\) and the electron is equal to the Sun -Earth distance. For an electron on the surface of Pb nucleus, find the ratio of the electric and gravitational potentials.	kapoor	22-04-18 19:04:00	n
[QUE/EPP-01021] Node id: 2658page Show that if deuterons are scattered by protons the maximum scattering angle in the centre of mass and the lab frames are \(120^o\) and \(30^o\) respectively. but if the protons are scattered by deuterons the maximum scattering angle is \(180^o\) in both the systems.	kapoor	22-04-18 19:04:07	n
[QUE/EPP-01029] Node id: 2667page If in a three body decay \[ A \longrightarrow B_1 + B_2 + B_3\] the energy momenta of the decay products are \(( E^i, \vec{p}^i), i=1,..,3\), show that $E_3$ is maximum when \[ ( p_1+p_2)^2 = (m_1+m_2)^2 \] Use Lagrange multipliers. Interpret this result geometrically. Generalize the above result to \(n\) particles.	kapoor	22-04-18 19:04:06	n
[QUE/EPP-01012] Node id: 2637page A four vector \((A^0, \vec{A})\) is called time like if \(\|A^0\| > \|\vec{A}\|\); space like if \(\|\vec{A}\| > \|A^0\| \); light like if \(\|\vec{A}\| = \|A^0\| \). Give an argument to show that the property of being time like, space like, or light like does not depend on the choice of the Lorentz frame.	kapoor	22-04-18 19:04:09	n
[QUE/EPP-01011] Node id: 2636page Show that a zero mass particle, such as photon, cannnot decay into two or more massive particles. Thus the process \begin{equation} \gamma \longrightarrow e^+ + e^- \end{equation} is forbidden for free photons in vacuum.	kapoor	22-04-18 19:04:19	n

Consider electron proton scattering at a given kinetic energy $T_L$ of the electron in the laboratory frame. Find the accessible energy range of $t$, the four momentum transfer squared.

Repeat the problem for the case of a proton beam with an electron target. Use the uncertainty relation to estimate the distance probed by the electron.Estimate what energy electrons would be required to probe down to 0.1 F

Show that the sign of time component of a time like vector remains the same in all Lorentz frames. Is this statement true for space like vectors?

If $S^\mu$ is a space like vector show that there exists a frame in which the time component $S^0$ vanishes. What is the corresponding statement for a time like vector?

If $A^\mu$ is a time like four vector show that there exist three independent vectors $B^\mu$ such that $B^\mu A_\mu=0$ and prove that all the three vectors are necessarily space like.

Extend the above result to the case when $A^\mu$ is (i) space like; (ii) light like.

The polarization vectors of photon with four momentum $k$, $\vec{\varepsilon}(\vec{k},\lambda), \lambda=1,2$, satisfy \[ \vec{k}\cdot \vec{\varepsilon}(k, \lambda) = 0 \] Taking $\eta_\mu=(1,0,0,0),$ and \[\varepsilon_{\mu}(\vec{k},\lambda)= (0,\vec{\varepsilon(k,\lambda)}), \quad \lambda=1,2, \] find a four vector $B^\mu$ such that it is orthogonal to $\varepsilon(k, \lambda)$ and $\eta_\mu $. Hint: What should the required vector $B_\mu$ be? Light like, time like or space like?
Use the above result to show that \begin{eqnarray} \sum_{\lambda=1}^2 \varepsilon_\mu(k,\lambda) \varepsilon_\nu(k,\lambda) &=& - g_{\mu\nu} + \eta_\mu\eta_\nu - \frac{k_\mu k_\nu}{(k\cdot\eta)^2-k^2} \nonumber\\ &\qquad &+ \frac{(k\cdot\eta)(\eta_\mu k_\nu+ k_\mu\eta_\nu)}{(k\cdot\eta)^2-k^2} - \frac{(k\cdot\eta)^2\eta_\mu\eta_\nu}{(k\cdot\eta)^2-k^2}.\nonumber \end{eqnarray}

If $A^\mu, B^\mu$ are time like, what can you say about the four vector $ A^\mu + B^\mu $? Will it always be time like or not?

    $Z^0$ particles are to be produced in $\bar{p} p$ collisions. What is the minimum energy required for
    (a) proton and for the anti-protons in c.m. frame ,
    (b) for anti proton in lab frame ( proton at rest)?

For each of the following decays, compute the momenta of the decay products in the rest frame of the decaying particle.

$ Z^0 \longrightarrow e^+\quad + \quad e^-$
$ W^+ \longrightarrow e^+\quad + \quad \nu$
$ \Sigma \longrightarrow N \quad+ \quad\pi $

Google and search internet to know the masses of various particles in the three decays.

If $A_{\lambda\sigma}$ and $B_{\lambda\sigma}$ are two second rank tensors satisfying \[ A_{\lambda\sigma}x^\lambda x^\sigma = B_{\lambda\sigma}x^\lambda x^\sigma\] for arbitrary four vector $x^\mu$, what is the relationship between $A_{\lambda\sigma}$ and $ B^{\lambda\sigma}$?

If $A^\mu$ and $B^\mu$ are light like four vector show that $B^\mu A_\mu=0$ if and only if \[ A^\mu = \text{constant} \times B^\mu\]

If $A^\mu$ is a light like four vector show that there exist a light like and two time like vectors $B^\mu$ such that $B^\mu A_\mu=0$. Extend the above result to the case when $A^\mu$ is
(a) space like; (b) light like.

Let $\epsilon$ and $\lambda$ be two light like fur vectors. What is the range of allowed values of $\epsilon\cdot\lambda$? Express your answer in terms of $\epsilon^0$ and $\lambda^0$.

Consider $S_{\mu\nu}$ to be a second rank tensor having the following properties:

$S_{\mu\nu}$ is symmetric : $ S_{\mu\nu} = S_{\nu\mu} $;
$S_{\mu\nu}$ is antisymmetric : $ S_{\mu\nu} = - S_{\nu\mu} $;
$S$ is traceless:$ g^{\mu\nu} S_{\mu\nu} = 0$.

Show that the above properties are remain invariant under Lorentz transformations.

If $p^\mu=(p^0,0,0,p)$ with $ p^{02} -\vec{p}^2 = m^2$, find all vectors $\epsilon^\mu$ such that $\epsilon^\mu p_\mu=0$. How many linearly independent vectors can you make?

Find the charge $Q$ which will exert the same force on an electron as Sun's gravitational pull on the earth when the distance between the charge $Q$ and the electron is equal to the Sun -Earth distance.
For an electron on the surface of Pb nucleus, find the ratio of the electric and gravitational potentials.

Show that if deuterons are scattered by protons the maximum scattering angle in the centre of mass and the lab frames are $120^o$ and $30^o$ respectively. but if the protons are scattered by deuterons the maximum scattering angle is $180^o$ in both the systems.

If in a three body decay \[ A \longrightarrow B_1 + B_2 + B_3\] the energy momenta of the decay products are $( E^i, \vec{p}^i), i=1,..,3$, show that $E_3$ is maximum when \[ ( p_1+p_2)^2 = (m_1+m_2)^2 \] Use Lagrange multipliers. Interpret this result geometrically. Generalize the above result to $n$ particles.

A four vector $(A^0, \vec{A})$ is called

time like if $|A^0| > |\vec{A}|$;
space like if $|\vec{A}| > |A^0| $;
light like if $|\vec{A}| = |A^0| $.

Give an argument to show that the property of being time like, space like, or
light like does not depend on the choice of the Lorentz frame.

Show that a zero mass particle, such as photon, cannnot decay into two or more massive particles. Thus the process \begin{equation} \gamma \longrightarrow e^+ + e^- \end{equation} is forbidden for free photons in vacuum.

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[QUE/EPP-01018]

[QUE/EPP-01020]

[QUE/EPP-01025]

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[QUE/EPP-01027]

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