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Title+Summary	Name	Date
[QUE/CM-05004] Node id: 2873page Sketch the equivalent one dimensional potential for the central force corresponding to \(F=-\frac{3a}{r^4}\) Draw the value of \(E^\prime\) such that a bound orbit is possible, state the conditions for the bound orbit	kapoor	22-04-08 08:04:53	n
[QUE/CM-05003] Node id: 2874page Show that the motion of a particle in the potential field \(V(r)=-\frac{k}{r} +\frac{h}{r^2}\) is the same as that of the motion under the Kepler potential alone, when expressed in terms of a coordinate system rotating or precessing around the center of force.	kapoor	22-04-08 08:04:10	n
[QUE/CM-05002] Node id: 2875page Sketch the equivalent one dimensional potential for the central force corresponding to \(F=-Br\). Draw the value of \(E^\prime\) such that a bound orbit is possible, state the conditions for the bound orbit (if any).	kapoor	22-04-08 08:04:52	n
[QUE/CM-06008] Node id: 2877page Consider a beam of alpha particles incident on a nucleus of atomic number \(Z\) and that the nucleus is a sphere of radius \(R\). Assume that the charge \(Ze\) of the nucleus is uniformly distributed inside the nucleus. What should be the energy, \(E_b\) of an alpha particle in the beam having an impact parameter \(b\). so that it just touches the surface of the nucleus, i.e. distance of minimum approach becomes equal to \(R\)? Find the corresponding scattering angle \(\theta(b,E_b)\) using the formula derived in the class. How do you expect your answers to change for alpha particle with different values of impact parameter \(b\). Consider situations (i) \(b<R\), (ii) \(b \approx R\) (iii) \( b> R\). What do you expect to happen to the alpha particle if(i) if it energy is less than \(E_b\); (ii) if it energy is greater than \(E_b\). Will the alpha particle get scattered at large angles? or, is the scattering angle expected to be small?	kapoor	22-04-08 08:04:57	n
[QUE/CM-06007] Node id: 2879page Consider a beam of alpha particles incident on a nucleus of atomic number \(Z\) and that the nucleus is a sphere of radius \(R\). Assume that the charge \(Ze\) of the nucleus is uniformly distributed inside the nucleus. What should be energy, \(E_0\) of an alpha particles moving directly towards the center of the nucleus, just touches the surface of the nucleus and recoils back? What do you expect to happen to the alpha particle if (i) if it energy is less than \(E_0\); (ii) if it energy is greater than \(E_0\). Will the alpha particle recoil back or go in the forward direction?	kapoor	22-04-08 08:04:05	n
[QUE/CM-06006] Node id: 2880page Find the scattering cross for scattering from hard sphere of radius \(R\) and show that the total cross section is \(\pi R^2\).	kapoor	22-04-08 08:04:55	n
[QUE/CM-06005] Node id: 2881page The neutron proton cross section is independent of the scattering angle in the lab frame for low energies. Find the dependence of the cross section on the scattering angle in the laboratory frame.	kapoor	22-04-08 08:04:11	n
[QUE/CM-06004] Node id: 2882page A central force potential frequently encountered in nuclear physics is the so called rectangular well, defined by the potential $$ V(r) = \left\{ \begin{array}{cr} 0 & r\ge a \qquad \\ -V_0 & r\le a \end{array} \right.$$ Show that the scattering produced by such a potential in classical mechanics is identical with the refraction of light rays by a sphere of radius $a$ and relative index of refraction $$ n= \sqrt{E+V_0 \over E } $$ Show that the differential cross section is $$ \sigma(\Theta) ={ n^2 a^2 \over 4\cos^2} {\Theta \over 2} \times { \left(n \cos{\Theta \over 2}-1 \right) \left(n - \cos{\Theta \over 2} \right)\over \left( 1+n^2 -2n \cos{\Theta \over 2} \right)^2}$$ Goldstein	kapoor	22-04-08 08:04:18	n
[QUE/CM-06003] Node id: 2883page Verify that the total cross for scattering for force $f= k r^{-3}$ is infinite. However if the force vanishes for $r > r_0$, show that there is some miminum scattering angle $\theta_0$; express the result in terms of $m, r_0$, and $k$ and show that the total cross section is $$\sigma_t = 2\pi \int_{\theta_0}^\pi \sigma(\Theta)\ d\,\Theta = \pi r_0^2 $$	kapoor	22-04-08 08:04:32	n
[QUE/CM-06002] Node id: 2884page Examine the scattering produced by a repulsive central force $f= k r^{-3}$. Show that the differential cross section is given by $$\sigma(\Theta)\ d\Theta ={k\over 2E} {(1-x) dx \over x^2 (2-x)^2 \sin \pi x} $$ where $x$ is the ratio ${\theta/ \pi}$ and $E$ is the energy. Goldstein	kapoor	22-04-08 08:04:51	n
[QUE/CM-06001] Node id: 2885page Show that if deuterons are scattered by protons the maximum scattering angle in the center-of-mass and laboratory frames are \(120^o\) and \(30^o\). respectively, but if the protons are scattered by deuterons, the maximum angle is \(180^o\) in both systems. Segre	kapoor	22-04-08 08:04:58	n
[QUE/QFT-15011] Rutherford Scattering in Second quantized Schrodinger equation Node id: 2963page Write the Lagrangian for a nonrelativistic particle moving in a potential \(V(r)\). Using the second quantized formalism compute the scattering cross section in the lowest order in the potential. Show that this result is same as the first Born approximation expression in nonrelativistic, first quantized formulation of quantum mechanics. Taking V(r) as Coulomb potential obtain the Rutherford formula:\[\left(\frac{d\sigma}{d\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\]	kapoor	22-04-08 08:04:37	n
[QUE/ME-02006] Rotation of Coordinate System Node id: 3195page \(\newcommand{\Prime}{^\prime}\) Let \({\bf A, B,..}\) be vector objects with components written as \(\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)\). Introduce \(\vec{A}\Prime=(A_1\Prime,A_2\Prime,A_3\Prime), \vec{B}\Prime=(B_1\Prime,B_2\Prime,B_3\Prime)\) etc. by means of equation \begin{equation}\label{EQ01} \vec{A}\Prime = \vec{A} -\sin \alpha (\hat{n}\times\vec{A}) + (1-\cos\alpha) \hat{n}\times (\hat{n}\times\vec{A}). \end{equation} and with similar equations for other vectors. Using vector identities show that \(\vec{A}\Prime\cdot\vec{B}\Prime=\vec{A}\cdot\vec{B}\); If \(\vec{C}=\vec{A}\times\vec{B}\), then \(\vec{C}\Prime\) is given by an equation similar to \eqref{EQ01}. How is is the expression \( \vec{A}\Prime\cdot(\vec{B}\Prime\times\vec{C}\Prime)\) related to \( \vec{A}\cdot (\vec{B}\times\vec{C}) \)?	kapoor	22-04-08 08:04:35	n
[QUE/ME-02001] Infnitesimal rotation of a vector Node id: 3196page Using geometrical arguments, show under an infinitesimal rotation of a vector \(\vec{A}\) about an axis \(\hat{n}\), and with suitable sign convention, the change in the vector is given by \(\Delta \vec{A}= -\Delta\theta \hat{n}\times{\vec{A}}\), where \(\Delta \theta\) is an infinitesimal angle of rotation.	kapoor	22-04-08 08:04:32	n
[QUE/ME-02002] Node id: 3197page \(\newcommand{\Prime}{^\prime}\)The coordinates of a point under rotation change as \begin{equation} \begin{pmatrix} x_1\Prime\\ x_2\Prime\\ x_3\Prime\end{pmatrix} = R \begin{pmatrix} x_1\\ x_2\\ x_3\end{pmatrix} .\end{equation} Show that the property that lengths and angles do not change under a rotation imples that \(R\) must be an orthogonal matrix. Hence show that \(\det R\) must be \(\pm 1\). What condition on the transformation will lead to the requirement that \(\det R\) must be equal to 1 and \(\det R=-1\) will ruled out?	kapoor	22-04-08 08:04:44	n
[QUE/ME-02003] Node id: 3198page Show that the rotation matrix for rotation by an angle \(\pi/3\) about the direction \((1,1,1)\) as the axis of rotation is given by \begin{equation} R = \frac{1}{3}\begin{pmatrix} 2 & 2 & -1\\ -1& 2 & 2 \\2 & -1 & 2 \end{pmatrix} \end{equation}	kapoor	22-04-08 08:04:58	n
[QUE/ME-02004] Node id: 3199page Find the rotation matrix for a rotation by an angle \(\cos^{-1}(3/5)\) about axis \((2,1,2)\). Answer: The rotation matrix is given by \[ R= \begin{pmatrix} 7/9 & -(4/9)& 4/9 \\[2mm] 28/45 & 29/45 & -(4/9)\\[2mm] -(4/45)& 28/45 & 7/9\end{pmatrix}.\]	kapoor	22-04-08 08:04:08	n
[QUE/ME-02005] Node id: 3200page Using the definition of Levi-Civita and Kronecker delta symbols to prove the following identity \begin{eqnarray} \epsilon_{i\,j\,k}\,\epsilon_{l\,m\,n}\ &=&\ \begin{vmatrix} \delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn} \end{vmatrix} \end{eqnarray}	kapoor	22-04-08 08:04:22	n
[QUE/ME-02007] Node id: 3201page \(\newcommand{\Prime}{^\prime}\)Let \({\bf A, B,..}\) be objects with components written as \(\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)\). Introduce \(\vec{A}\Prime=(A_1\Prime,A_2\Prime,A_3\Prime), \vec{B}\Prime=(B_1\Prime,B_2\Prime,B_3\Prime)\) etc. by means of equation \begin{equation}\label{EQ01} \begin{pmatrix}A_1\Prime\\A_2\Prime\\A_3\Prime\end{pmatrix} = R \begin{pmatrix}A_1\\A_2\\A_3\end{pmatrix} \end{equation} and with similar equations for other vectors. Using the fact that the matrix \(R\) is orthodgonal matrix show that \({A}_i\Prime\vec{B}_i\Prime=A_i B_i\); If \(C_i\Prime=\epsilon_{ijk}A_j\Prime B_k\Prime\), then \(\vec{C}\Prime\) is given by an equation similar to \eqRef{EQ01}, where, of course, \(C_i=\epsilon_{ijk}A_j B_k\) The value of \( \vec{A}\Prime\cdot(\vec{B}\Prime\times\vec{C}\Prime)\) is independent of the matrix \(R\), if \(\det R=1\). What happens if \(\det R=-1\)?	kapoor	22-04-08 08:04:33	n
[QUE/ME-02008] Node id: 3202page \(\newcommand{\Prime}{^\prime}\)For a four vector \(x=(\vec{x},x_4)\equiv(x_1,x_2,x_3,x_4)\), define a \(2\times2\) matrix \(M\) by \[M = x_\mu\sigma_\mu= x_4 + \vec{x}\cdot\vec{\sigma}\] where \(\sigma_4\) is \(2\times2\) identity matrix and \(\vec{\sigma}\) are Pauli matrices given by \begin{equation} \sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \sigma_1=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad \sigma_1=\begin{pmatrix}1&0\\0&-1\end{pmatrix} \end{equation} Also define a matrix \(U\) by \begin{equation} U = \cos\frac{\alpha}{2} + i\sin\frac{\alpha}{2}(\hat{n}\cdot\vec{\sigma}) \end{equation} where \(\hat{n}=(n_1,n_2,n_3)\) is a unit vector. Show that \(U\) is unitary and \(\det U=1\). Prove that \(x_\mu = \frac{1}{2} Tr(\sigma_\mu M)\) Let \(M\Prime= U M U^\dagger\) Compute \(x_\mu\Prime \) and show that \begin{equation} x_4\Prime=x_4; \vec{x}\Prime=\vec{x}-\sin\alpha(\hat{n}\times\vec{x}) +(1-\cos\alpha)\hat{n}\times(\hat{n} \times\vec{x}) \end{equation} In Newtonian mechanics,\(x_4\) can be identified with time. Comment The last equation shows that to every rotation in three dimensions there are two \(SU(2)\) matrices given by \(\pm U\).	kapoor	22-04-08 08:04:40	n

Sketch the equivalent one dimensional potential for the central force corresponding to $F=-\frac{3a}{r^4}$ Draw the value of $E^\prime$ such that a bound orbit is possible, state the conditions for the bound orbit

Show that the motion of a particle in the potential field $V(r)=-\frac{k}{r} +\frac{h}{r^2}$ is the same as that of the motion under the Kepler potential alone, when expressed in terms of a coordinate system rotating or precessing around the center of force.

Sketch the equivalent one dimensional potential for the central force corresponding to $F=-Br$. Draw the value of $E^\prime$ such that a bound orbit is possible, state the conditions for the bound orbit (if any).

Consider a beam of alpha particles incident on a nucleus of atomic number $Z$ and that the nucleus is a sphere of radius $R$. Assume that the charge $Ze$ of the nucleus is uniformly distributed inside the nucleus. What should be the energy, $E_b$ of an alpha particle in the beam having an impact parameter $b$. so that it just touches the surface of the nucleus, i.e. distance of minimum approach becomes equal to $R$? Find the corresponding scattering angle $\theta(b,E_b)$ using the formula derived in the class.
How do you expect your answers to change for alpha particle with different values of impact parameter $b$. Consider situations (i) $b<R$, (ii) $b \approx R$ (iii) $ b> R$.
What do you expect to happen to the alpha particle if(i) if it energy is less than $E_b$; (ii) if it energy is greater than $E_b$. Will the alpha particle get scattered at large angles? or, is the scattering angle expected to be small?

Consider a beam of alpha particles incident on a nucleus of atomic number $Z$ and that the nucleus is a sphere of radius $R$. Assume that the charge $Ze$ of the nucleus is uniformly distributed inside the nucleus. What should be energy, $E_0$ of an alpha particles moving directly towards the center of the nucleus, just touches the surface of the nucleus and recoils back?
What do you expect to happen to the alpha particle if (i) if it energy is less than $E_0$; (ii) if it energy is greater than $E_0$. Will the alpha particle recoil back or go in the forward direction?

Find the scattering cross for scattering from hard sphere of radius $R$ and show that the total cross section is $\pi R^2$.

The neutron proton cross section is independent of the scattering angle in the lab frame for low energies. Find the dependence of the cross section on the scattering angle in the laboratory frame.

A central force potential frequently encountered in nuclear physics is the so called rectangular well, defined by the potential $$ V(r) = \left\{ \begin{array}{cr} 0 & r\ge a \qquad \\ -V_0 & r\le a \end{array} \right.$$
Show that the scattering produced by such a potential in classical mechanics is identical with the refraction of light rays by a sphere of radius $a$ and relative index of refraction $$ n= \sqrt{E+V_0 \over E } $$ Show that the differential cross section is $$ \sigma(\Theta) ={ n^2 a^2 \over 4\cos^2} {\Theta \over 2} \times { \left(n \cos{\Theta \over 2}-1 \right) \left(n - \cos{\Theta \over 2} \right)\over \left( 1+n^2 -2n \cos{\Theta \over 2} \right)^2}$$

Goldstein

Verify that the total cross for scattering for force $f= k r^{-3}$ is infinite. However if the force vanishes for $r > r_0$, show that there is some miminum scattering angle $\theta_0$; express the result in terms of $m, r_0$, and $k$ and show that the total cross section is $$\sigma_t = 2\pi \int_{\theta_0}^\pi \sigma(\Theta)\ d\,\Theta = \pi r_0^2 $$

Examine the scattering produced by a repulsive central force $f= k r^{-3}$. Show that the differential cross section is given by $$\sigma(\Theta)\ d\Theta ={k\over 2E} {(1-x) dx \over x^2 (2-x)^2 \sin \pi x} $$ where $x$ is the ratio ${\theta/ \pi}$ and $E$ is the energy.

Goldstein

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

Segre

Write the Lagrangian for a nonrelativistic particle moving in a potential $V(r)$. Using the second quantized formalism compute the scattering cross section in the lowest order in the potential. Show that this result is same as the first Born approximation expression in nonrelativistic, first quantized formulation of quantum mechanics. Taking V(r) as Coulomb potential obtain the Rutherford formula:\[\left(\frac{d\sigma}{d\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\]

$\newcommand{\Prime}{^\prime}$
Let ${\bf A, B,..}$ be vector objects with components written as $\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)$. Introduce $\vec{A}\Prime=(A_1\Prime,A_2\Prime,A_3\Prime), \vec{B}\Prime=(B_1\Prime,B_2\Prime,B_3\Prime)$ etc. by means of equation \begin{equation}\label{EQ01} \vec{A}\Prime = \vec{A} -\sin \alpha (\hat{n}\times\vec{A}) + (1-\cos\alpha) \hat{n}\times (\hat{n}\times\vec{A}). \end{equation} and with similar equations for other vectors.

Using vector identities show that

$\vec{A}\Prime\cdot\vec{B}\Prime=\vec{A}\cdot\vec{B}$;
If $\vec{C}=\vec{A}\times\vec{B}$, then $\vec{C}\Prime$ is given by an equation similar to \eqref{EQ01}.

How is is the expression $ \vec{A}\Prime\cdot(\vec{B}\Prime\times\vec{C}\Prime)$ related to $ \vec{A}\cdot (\vec{B}\times\vec{C}) $?

Using geometrical arguments, show under an infinitesimal rotation of a vector $\vec{A}$ about an axis $\hat{n}$, and with suitable sign convention, the change in the vector is given by $\Delta \vec{A}= -\Delta\theta \hat{n}\times{\vec{A}}$, where $\Delta \theta$ is an infinitesimal angle of rotation.

$\newcommand{\Prime}{^\prime}$The coordinates of a point under rotation change as \begin{equation} \begin{pmatrix} x_1\Prime\\ x_2\Prime\\ x_3\Prime\end{pmatrix} = R \begin{pmatrix} x_1\\ x_2\\ x_3\end{pmatrix} .\end{equation} Show that the property that lengths and angles do not change under a rotation imples that $R$ must be an orthogonal matrix. Hence show that $\det R$ must be $\pm 1$.
What condition on the transformation will lead to the requirement that $\det R$ must be equal to 1 and $\det R=-1$ will ruled out?

Show that the rotation matrix for rotation by an angle $\pi/3$ about the direction $(1,1,1)$ as the axis of rotation is given by \begin{equation} R = \frac{1}{3}\begin{pmatrix} 2 & 2 & -1\\ -1& 2 & 2 \\2 & -1 & 2 \end{pmatrix} \end{equation}

Find the rotation matrix for a rotation by an angle $\cos^{-1}(3/5)$ about axis $(2,1,2)$.

Answer:
The rotation matrix is given by \[ R= \begin{pmatrix} 7/9 & -(4/9)& 4/9 \\[2mm] 28/45 & 29/45 & -(4/9)\\[2mm] -(4/45)& 28/45 & 7/9\end{pmatrix}.\]

Using the definition of Levi-Civita and Kronecker delta symbols to prove the following identity \begin{eqnarray} \epsilon_{i\,j\,k}\,\epsilon_{l\,m\,n}\ &=&\ \begin{vmatrix} \delta_{il}&\delta_{im}&\delta_{in}\\ \delta_{jl}&\delta_{jm}&\delta_{jn}\\ \delta_{kl}&\delta_{km}&\delta_{kn} \end{vmatrix} \end{eqnarray}

$\newcommand{\Prime}{^\prime}$Let ${\bf A, B,..}$ be objects with components written as $\vec{A}=(A_1,A_2,A_3), \vec{B}=(B_1,B_2,B_3)$. Introduce $\vec{A}\Prime=(A_1\Prime,A_2\Prime,A_3\Prime), \vec{B}\Prime=(B_1\Prime,B_2\Prime,B_3\Prime)$ etc. by means of equation \begin{equation}\label{EQ01} \begin{pmatrix}A_1\Prime\\A_2\Prime\\A_3\Prime\end{pmatrix} = R \begin{pmatrix}A_1\\A_2\\A_3\end{pmatrix} \end{equation} and with similar equations for other vectors.

Using the fact that the matrix $R$ is orthodgonal matrix show that

${A}_i\Prime\vec{B}_i\Prime=A_i B_i$;
If $C_i\Prime=\epsilon_{ijk}A_j\Prime B_k\Prime$, then $\vec{C}\Prime$ is given by an equation similar to \eqRef{EQ01}, where, of course, $C_i=\epsilon_{ijk}A_j B_k$
The value of $ \vec{A}\Prime\cdot(\vec{B}\Prime\times\vec{C}\Prime)$ is independent of the matrix $R$, if $\det R=1$. What happens if $\det R=-1$?

$\newcommand{\Prime}{^\prime}$For a four vector $x=(\vec{x},x_4)\equiv(x_1,x_2,x_3,x_4)$, define a $2\times2$ matrix $M$ by \[M = x_\mu\sigma_\mu= x_4 + \vec{x}\cdot\vec{\sigma}\] where $\sigma_4$ is $2\times2$ identity matrix and $\vec{\sigma}$ are Pauli matrices given by \begin{equation*} \sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad \sigma_1=\begin{pmatrix}0&-i\\i&0\end{pmatrix},\quad \sigma_1=\begin{pmatrix}1&0\\0&-1\end{pmatrix} \end{equation*} Also define a matrix $U$ by \begin{equation*} U = \cos\frac{\alpha}{2} + i\sin\frac{\alpha}{2}(\hat{n}\cdot\vec{\sigma}) \end{equation*} where $\hat{n}=(n_1,n_2,n_3)$ is a unit vector.

Show that $U$ is unitary and $\det U=1$.
Prove that $x_\mu = \frac{1}{2} Tr(\sigma_\mu M)$
Let $M\Prime= U M U^\dagger$ Compute $x_\mu\Prime $ and show that \begin{equation} x_4\Prime=x_4; \vec{x}\Prime=\vec{x}-\sin\alpha(\hat{n}\times\vec{x}) +(1-\cos\alpha)\hat{n}\times(\hat{n} \times\vec{x}) \end{equation}

In Newtonian mechanics,$x_4$ can be identified with time.

Comment The last equation shows that to every rotation in three dimensions there are two $SU(2)$ matrices given by $\pm U$.

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[QUE/QFT-15011] Rutherford Scattering in Second quantized Schrodinger equation

[QUE/ME-02006] Rotation of Coordinate System

[QUE/ME-02001] Infnitesimal rotation of a vector

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