05.01.24
Durga Prasad Sir, IOP
Intro : $\alpha, \beta, \gamma$ decay and detectors.
$p \rightarrow n$ free proton to neutron conversion doesn't happen due to mass restriction $m_n \gt m_p$, So, $n \rightarrow p \rightarrow H_2O$ happens and nature exists.
$e^- - e^+$ annihilation expt. in CMP using $^{22}Na$.
2 Photon $e^+ e^-$ angular momentum experiment.
Energetically favourable colliding beam experiment. $\sqrt{s} \propto E_a$.
Not fixed target $s\propto\sqrt{E_a}$
Anjan Giri, (IIT, Hyed)
Particle Physics (02:00 am)
Focus on neutrino and flavor Physics.
Sudhanwa Patra (HoD., Dept. Of Physics, IIT, Bhillai)
Fascinating neutrino Physics (03:30 pm)
06.01.24
Chandrasekhar Bhamidipati
General Relativity refresher course.
Newton's laws: $p=mv$, $F=dp/dt = dm/dt, v + m dv/dt = 0 + ma$ As mass is generally constant.
$F_i =m_i \ddot{x_i}$ where $i=1, 2, 3$ for (x, y, z). Newton's Laws are defined in the background of Euclidean space. The metric/line element of Euclidean space is $ds^2= dx^2 + dy^2 + dz^2$, $x'=x+a, y'=y+b, z'=z+c$ where a,b,c are constants.
$F'=F \Rightarrow \frac{d^2x'}{dt^2} =\frac{d^2x}{dt^2}$
1) These are translational symmetry.
2) Rotational symmetry.
Exp in 2d case
$$\begin{bmatrix}x^{\prime} \\ y^{\prime}\end{bmatrix} =
\begin{bmatrix}\cos\theta & \sin\theta \\ -\sin\theta & \cos\theta\end{bmatrix}
\begin{bmatrix}x \\ y\end{bmatrix}$$
$x'=x\cos\theta + y\sin \theta$ and $y'=-x\sin\theta+y\cos\theta$.
Newton's Laws are invariant, $F'=F$.
3) Galilean boosts.
( A transformation that mixes space and time is called a boost.)
O-rest, O' -> moving with velocity$v_0$.
Galilean boost: $x'=x-v_0t$, $y'=y$, $z'=z$, $t'=t$.
$F'= \frac{d^2x'}{dt^2} =\frac{d}{dt}(\frac{dx}{dt}-v_0) = \frac{d^2x}{dt^2} =F$
Newton's Laws are invariant.
* Maxwell's equations are NOT invariant under Galilean transformation.
* Gauss's law and Ampere's law are invariant.
Newton's Laws and Maxwell's equation are experimentally verified. So, are the Galilean
transformation valid always?
Qn t' =t?
Einstein:
A chalk falls form a height. Time taken by observer at Jajpur is t.
Time taken by observer at Mumbai is t'. Observer at Mumbai has to know
that the chalk has fallen. For t=t', the implicit assumption is that the signal must
travel instantaneously (Galinean transformation).
Einstein at 1900 questioned that. Michelson and morley found in an expt: speed of
light is constant irrespective of the motion of the observer.
Einstein takes the statement of Michelson & Morley experiment as a postulate .
$v_{max}=c$ Outcome : $t' \ne t$.
Gedanken
$T=2t'$
$$ c^2dt^2 =v^2dt^2+l^2$$
$$\Rightarrow dt^2 = \frac {l^2}{c^2-v^2}=\frac{l^2}{c^2} \frac{1}{1-v^2/c^2}$$
$$\Rightarrow dt = \frac{l/c}{\sqrt{1-v^2/c^2}} = \frac{l'}{c} \frac{1}{\sqrt{1-v^2/c^2}}
= \frac{cdt'}{c} \frac{1}{\sqrt{1-v^2/c^2}} = \frac{dt'}{\sqrt{1-v^2/c^2}}$$
$dt = \frac{dt'}{\sqrt{1-v^2/c^2}}$
So, $T=\gamma T'$(Time dilation) where, $\gamma = 1/\sqrt{1-v^2/c^2}$.
(Experiment done on the train.)
For experiment done on the ground, similar thought experiment can be done.
$$x'=\gamma (x-v_0t) \ y'=y \z'=z \ t'=\gamma t$$
CONSTANCY OF SPEED OF LIGHT
Ratio of distance to time has to be constant.
$c=c' \Rightarrow c=r/t = c' = r'/t'$
$\Rightarrow ct = r \Rightarrow c^2t^2 = r^2 \textsf{ OR } \textcolor{red}{ -c^2t^2+r^2=0}$
Similarly, $\textcolor{red} {-c^2t'^2+r'^2=0}$
$$-c^2t^2+r^2=0= -c^2t'^2+r'^2 \Rightarrow -c^2t^2+x^2+y^2+z^2=-c^2t'^2+x'^2+y'^2+z'^2$$
Mathematician Lorentz searched the most general transformations that leads to the
equations unchanged.
$x' =\gamma (x-v_0t) , y'=y , z'=z, , t'= \gamma ( t+\delta)$
$\delta \equiv f(c, v_0, x)$
We need to find $\gamma$ and $\delta$.
$$c^2t'^2=x'^2+y'^2+z'^2 \
\Rightarrow \gamma^2 (x^2-2vxt+v^2t^2)+y^2+z^2 = c^2\gamma^2 (t^2+2\delta t+\delta^2)$$
Equating the coefficients of $t$, we get $-2\gamma^2 vx = 2 c^2 \gamma^2 \delta t$
$$ \delta = -vx/c^2 \quad \gamma = \frac{1}{\sqrt{1-v^2/c^2}}$$
Lorentz Transformations
$$\textcolor{red}{x'= \gamma (x-v_0 t) \ y'=y \ z' = z \ t' = \gamma (t-vx/c^2) }$$
Why this transformation is called Lorentz transformation not Einstein transformation?
$$\begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix} =
\begin{pmatrix}\cos\theta & \sin\theta\\ -\sin\theta & \cos\theta\end{pmatrix}
\begin{pmatrix}x\\ y\end{pmatrix}$$
This transformation leads to
$$x'^2+y'^2=x^2+y^2.$$
Lorentz was interested in finding a transformation that leads to
$$-x'^2+y'^2=-x^2+y^2.$$
Lorentz found that the transformation is
$$\begin{pmatrix}x^{\prime} \\ y^{\prime}\end{pmatrix} =
\begin{pmatrix}\cosh\theta & \sinh\theta\\ \sinh\theta & \cosh\theta \end{pmatrix}
\begin{pmatrix}x\\ y\end{pmatrix}$$
Line element $ds^2=-c^2dt^2+dx^2+dy^2+dz^2$ . In Special theory of relativity there is
a possibility $ds^2\le 0$. Explanation of Einstein leads to 'Light Cone'.
In GTR, $ds^2=dx^2+dy^2+dz^2 \gt 0$.
Anjan Giri (11:00 am)
Nature's elementary building blocks.
In 1972, Kobayashi Maskawa advocated family of quarks to account CP violation.
Positron (an anti-particle) actually exists. It is used in medical diagnostic (PET scanning).
Neutrinos: Bananas emit a lots of neutrinos(1 mn/day). (Radioactive due to presence of K)
Hadron therapy: To treat cancerous cells using protons.
$$4 _0^1H +2e^- \rightarrow _2^4 He +\nu +h\nu$$
This reaction takes nearly $10^{15}$ s. So, we owe our existence to the weak interaction.
Otherwise, all energies must have formed instantly.
Sudhanwa Patra (02:00 pm)
Neutrino Analysis.
Two flavour neutrino oscillation.
Neutrino massless or mass << eV ; compare electron mass = 0.5 MeV.
Prof. Hiranmay Mishra, NISER, (03:40 pm).
Electromagnetism (David Tong)
Notes shared.
05.01.24
Prof. Hiranmay Mishra
Electrodynamics and Gauge.
Prof. Snigdha Mishra (Berhampur University)
Grand Unification Theory(GUT)
Chandrasekhar Bhamidipati (Contd.)
Invariant Interval $ds^2=ds'^2$.
With Lorentz transformation $ds^2 \gt 0$ or $ds^2 \le 0$ (light cone).
$v\lt\lt c \Rightarrow$ LT --> GT.
Velocity of particle in Special Relativity
$$ x'= \gamma (x-v_0 t) \Rightarrow dx' = \gamma (dx-v_0 dt)$$
$$ t'= \gamma (t-v_0 x/c^2) \Rightarrow dt' = \gamma (dt-v_0 dx/c^2)$$
$$ \frac{dx'}{dt'} = \frac{dx-v_0dt}{dt-v_0,dx/c^2} =
\frac{dt(\frac{dx}{dt}-v_0)}{dt(1-\frac{v_0}{c^2}\frac{dx}{dt})} $$
$$\mathbf{u'=\frac{u-v_0}{1-uv_0/c^2}}$$
Exp:
O frame Particle 0.8c -> O' frame <- 0.9c
$$u'= \frac{0.8c -(-0.9c)}{1-0.8c(-0.9c)/c^2}=0.99c$$
$$\mathbf{u_x'=\frac{u_x-v}{1-u_xv/c^2}}$$
$$\mathbf{u_y'=\frac{1}{\gamma} \frac{u_y}{1-u_xv/c^2}}$$
$$\mathbf{u_z'=\frac{1}{\gamma} \frac{u_z}{1-u_xv/c^2}}$$
Relativistic mass
Imposing energy and momentum conservation in STR
$$m(v)= m_0\gamma = \frac{m_0}{1-\frac{v^2}{c^2}}$$
Relativistic Energy
KE:
$$K= \int_{x_i}^{x_f} F, dx$$
$$F=dp/dt = mdv/dt + vdm/dt$$
$$m^2 = \frac{m_0^2}{1-v^2/c^2} \Rightarrow m^2(1-v^2/c^2)=m_0^2
\Rightarrow m^2c^2 -m^2 v^2 =m_0^2c^2 \
\Rightarrow 2mdm/dt c^2 -2mdm/dt v^2 -2m^2v dv/dt =0 \
\Rightarrow c^2 \frac{dm}{dt} =v^2\frac{dm}{dt} + mv \frac{dv}{dt} \
\Rightarrow \frac{c^2}{v} \frac{dm}{dt} = v\frac{dm}{dt} + m \frac{dv}{dt}$$
So,
$$K=\int_{x_i}^{x_f} \frac{c^2}{v} \frac{dm}{dt} dx =
\int_{x_i}^{x_f} \frac{c^2}{v} dm
\frac{dx}{dt} =
\int_{x_i}^{x_f} \frac{c^2}{v} v, dm = c^2(m_f-m_i)$$
Kinetic energy of a Relativistic particle is $K=mc^2-m_0c^2$.
$$mc^2 = K + m_0c^2$$
Total energy = Kinetic energy + Rest mass energy .
$E=mc^2$ is Relativistic Energy.
Relativistic Energy and momentum relation.
$$E^2= p^2c^2 + m_0^2c^4$$
For a photon; $m_0=0, , E^2=p^2c^2 \Rightarrow \bf{E=pc}$.
Line element in special relativity
$$ds^2=-c^2dt^2 +dx^2+dy^2+dz^2 \
y=z=0 \Rightarrow ds^2= -c^2 dt^2. + dx^2$$
$$ds^2=0 \Rightarrow-c^2dt^2+dx^2=0 \Rightarrow dx^2=c^2dt^2
\Rightarrow \frac{dx}{dt} =\pm c \Rightarrow v=c$$
$$\int dx = \int \pm c dt \Rightarrow x=\pm ct + a_0 $$
$a_0$ is a constant. Trajectory of a straight line.
$ds^2\lt 0 \Rightarrow-c^2dt^2+dx^2\lt 0$ Time part is more than space part. $v\lt \lt c$.
$ds^2\gt 0 \Rightarrow v\gt c$ .
Space-time around us is divided into three regions. 1. Space like region,
2. Time like region and 3. Light like region. Units used c=1, [L]=[T].
Light Cone:
Durga Prasad Mahapatra (IOP) (03:30 pm)
Detector Technology.
Book: Leo, Experimental Techniques.
09.01.24
Chandrasekhar Bhamidipati (10:00 am)
GT Relativity
Lorentz Boost.
Prof Snigdha Mishra contd. 11:30 am
Prof. H. Mishra contd. 02:00 pm
Prof. Prashant Kumar Panigrahi IISER KOL, ITER (03:30 pm)
Quantum Computation.
IBM gives free account for quantum computing.
QRNG selling for Rs 10,000 using quantum computation.
10.01.24
Sudhanwa Patra
Raghunath Sahoo
Quantum Computing contd.
11.01.24
12.01.24
Prof. Rukmani Mohanta, Hyderabad University
Heavy flavour Physics and CP violation.
Prof. Raghunath Sahoo.
13.01.24
Pravata Mohanty (TIFR)
pkm@tifr.res.in
Introduction to cosmic rays.