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Prove that the free particle solutions of Klein Gordon equation \(f_q(x)\), given by
\[ f_q(x) = \frac{1}{\sqrt{(2\pi)^3}} e^{-iqx},\]  obey the orthononality relations \[ \begin{eqnarray}   i \int d^3x f_q^*{x} \{\partial_0 f_p(x)\} - \{\partial _0 f_q(x)\} f_p^*(x) = 2\omega_q\delta(\vec{q}-\vec{p}). \end{eqnarray} \] and find the value of \( \int d^3x \big[f_q(x) (f_q(x) \{\partial_0 f_p(x)\} - \{\partial _0 f_q(x)\} f_p(x))\big]\)
                    

Starting from the Lagrangian for a complex scalar field obtain the Hamiltonian for a free complex Klein Gordon field.
Write ETCR and for the quantized field prove that \begin{equation}\label{EQ01}
\big[H, \pi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) .
\end{equation} Does relation,\eqref{EQ01}, hold only as equal time commutator or does it hold for  \(H\) and \(\pi(x)\) at arbitrary different times ?  Explain your answer.
Use \eqref{EQ01} to derive the usual Euler Lagrange equation of motion for the Klein Gordon field.

   Prove that
  \(\begin{equation}
    \int_{t_0}^ t \, dt_1 \int_{t_0}^ {t_1} dt_2\, H^\prime_I(t_1)
      H^\prime_I(t_2) = \frac{1}{2}      \int_{t_0}^ t \, dt_1 \int_{t_0}^ t \,
       dt_2  T\big( H^\prime_I(t_1) H^\prime_I(t_2)\big)
       \end{equation}\)

Write Dirac equation in external electromagentic potential  \(A_\mu(\vec{x})\). Assume the potentials for Coulomb interactions with a nucleus of charge \(Ze\) to be
\[ \vec{A}=0, \quad A_0 = \frac{Ze}{4\pi |\vec{x}|}.  \]
Show that the differential cross section for electron scattering from nucleus is
given by
\[\frac{d\sigma}{d\Omega}= \left(\frac{d\sigma}{d\Omega}\right)_\text{R} \Big(1-v^2\sin^2(\theta/2)\Big)\] where
\[\left(\frac{d\sigma}{d\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\]
is the Rutherford cross section for nonrelativistic Coulomb scattering.

A sprinkler wagon wets asphalt on a hot summer day. the power of the mottor is barely great enough to over the combined friction of the ground and the wheels, of the air, and the axle bearings. The vehicle therefore behaves  as if under no forces. Let $m$ be the mass of the water in the water tank at any instant plus the constant mass of the empty vehicle. Let the amount of water  squirted out per unit time $\mu = -\dot{m}$ , be its exit velocity towards rear, $q$, a  seen from the wagon, or $v-q$  as seen from the street, $v$ be the speed of the vehicle. Then using Newton's second law the rate of change of momentum og the   water + wagon = Force on the wagon. or,      \begin{eqnarray}
          \dot{p} = 0 \\
          \frac{d}{dt}(m v) =0\\
          \dot{m} v + m \frac{dv}{dt} =0 \\
          or, m \frac{dv}{dt}= \mu v
\end{eqnarray} It would then appear that the acceleration of the wagon is independent of the
 exit velociy $q$. This is paradoxical. Give your comments.

Source:Sommerfeld

\(\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}
\)
For a real free Dirac field, mass \(m\),  compute \[ \matrixelement{0}{\psi(x)\psi(y)}{\vec{p},r; \vec{q},s}\] and show that the result is  properly anti-symmetrized wave function for two identical fermions with momenta \(\vec{p},r;\vec{q},s\) and  spins \(r,s\). Here \(\ket{\vec{k}, \vec{q}}\) is the state with two fermions with momenta \(\vec{p}, \vec{q}\) and spins \(r,s\).

Let \(\Gamma\) denote a product of Dirac matrices. Define \(\widetilde{\Gamma}\) by \begin{equation*} \bar{u}(s,q) \widetilde{\Gamma}u(r,p) = (\bar{u}(r,p)^\dagger\Gamma u(s,q))^* . \end{equation*} Use the above definition of \(\widetilde{\Gamma}\) and show that

1. \(\widetilde{\Gamma} = \gamma_0 \Gamma^\dagger \gamma_0. \)
2.      \(\widetilde{\gamma^\mu} = \gamma^\mu \)
3.     \(\widetilde{\gamma_5} = -\gamma_5\)
4.     \(\widetilde{\gamma^\mu \gamma_5}= - \gamma^\mu \gamma_5\)

Compute \(\widetilde{\sigma}_{0k}\) and \(\widetilde{\sigma}_{ij}\) and hence show      that \(\widetilde{\sigma}_{\mu\nu}=\sigma_{\mu\nu}\)

 Express the field momentum  \begin{equation*} P^k = \int d^3x \big(  \pi(x) \partial^k \phi(x) +  \pi^*(x)\partial^k \phi(x)^*\big)     \end{equation*}    in terms of creation and annihilation operators.

An empty freight car, which is travelling along a horizontal track, passes under funnel filled with sand as shown in the diagram. The sand is released into the car at the rate  of $\dot{m}$. Find the force needed to keep the car at constant speed $v_0$.

Source{HSM 67}.