Browse & Filter

 The values of \(mc^2\) for the pion \(\pi^+\) and muon \(\mu^+\) are 139.57MeV and 105.66 MeV respectively. Find the kinetic energy of the muon decay in \[ \pi^+ \longrightarrow \mu^+  + \nu_\mu \] assuming that the neutrino is massless. For a neutrino of finite but very small mass \(m_\nu\) show that, compared with the case of a massless neutrino, the muon momentum would be reduced by the fraction \begin{equation*}    \frac{\Delta p}{p} = - \frac{m_\nu^2(m_\pi^2+m_\mu^2)}{(m_\pi^2-m_\mu^2)^2} ~\text{MeV} \approx -\frac{m_\nu^2}{10^4}. \end{equation*} where \(m_\nu\) is neutrino mass in MeV.

An analysis of measurement of position, the famous  Heisenberg microscope thought experiment, leads to  uncertainty relation with momentum uncertainty identified with the momentum transfer. Using similar idea show that the distance probed in a scattering experiment  \[  A + B \longrightarrow A + B\] is approximately  \(1/\sqrt{t}\), where $t=(p_A-q_A)^2\), where \(p_A, p_B\) denote the initial and \(q_A, q_B\) denote the final four momenta of the two particles.

Find the charge \(Q\) that will exert the same force on an electron at 1 nm as the Sun's gravitational pull on the Earth.

     For a particle moving in spherically symmetric potential             $$V(r) = -V_0 \exp(-r/r_0)$$       and  having the wave function               $$\psi(r) = N \exp(-\alpha r/r_0) $$       show that        $$\langle \mbox{ K.E. } \rangle = {\hbar^2\alpha^2\over 2mr_0^2} ;
         \qquad \langle\,V(r)\,\rangle = -{8V_0 \alpha^3\over (2\alpha +1)^3}$$

   For a particle having the wave function           $$\psi(x) = N
\exp(-x^2/\alpha^2)$$       compute the averages of the following dynamical
variables.
          (a) kinetic energy,
          (b) $V_1(x) = V_0 |x|^{2m+1}$
          (c)  $V_2(x) =kx^2$

For a harmonic oscillator in the ground state find the average values of
kinetic energy, potential energy and $|x|^{2m+1}.$

Given that :The vector space needed to describe a particular physical system is two
dimensional complex vector space. The  states are therefore represented by 2
component complex column vectors. The observables for this system are $2\times2$
matrices. A set of three dynamical variables of the system, $X,Y,Z,$ are be
represented by $2\times 2$ matrices denoted by $\sigma_x, \sigma_y,\sigma_z$,
where $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$ Answer the following question
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Question :      What vector would represent the state of the system if it is known that
the system has definite value $+1$ for the dynamical variable $X$? What vector
would represent the state if the system has definite value $-1$ for the variable
$Y$.

 The vector space needed to describe a particular physical system is two
dimensional complex vector space. The  states are therefore represented by 2
component complex column vectors. The observables for this system are $2\times2$
matrices. A set of three dynamical variables of the system, $X,Y,Z,$ are be
represented by $2\times 2$ matrices denoted by $\sigma_x, \sigma_y,\sigma_z$,
where  $$  \sigma_x
=\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad
\sigma_y=\begin{pmatrix} 0&-i\\i&0\end{pmatrix}, \qquad
\sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$     What values are
experimentally allowed if one measures the dynamical variable

1.        $X$ 
2.       $Z$ 
3.       $T= X^2+Y^2+Z^2$

The vector space needed to describe a particular physical system is two dimensional complex vector space. The  states are therefore represented by 2 component complex column vectors. The observables for this system are $2\times2$
matrices. A set of three dynamical variables of the system, $X,Y,Z,$ are be represented by $2\times 2$ matrices denoted by $\sigma_x, \sigma_y,\sigma_z$, where $$  \sigma_x =\begin{pmatrix}0&1\\1&0\end{pmatrix},\qquad \sigma_y=\begin{pmatrix}0&-i\\i&0\end{pmatrix}, \qquad \sigma_z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}.$$  If the state vector of the system is given by $$   \left( \begin{array}{r} \displaystyle{1\over\sqrt{5}}\\[4mm] \displaystyle {-2\over\sqrt{5}} \end{array} \right) $$ Find the probability that

1. a measurement of $X$ will give value $1$.
2. a measurement of $Y$ will give value $-1$.
3. a measurement of $Z$ will give value $1$.
4. a measurement of $X+Y$ will give value $\sqrt{2}$.