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Let \(\chi,\phi, \psi\)  be a set of representative vectors of three unit rays in Hilbert space and let  \(\chi',\phi', \psi'\) be another set of representatives of the same rays. Prove that \[(\chi,\phi)(\phi, \psi)(\psi, \chi) = (\chi',\phi')(\phi', \psi')(\psi', \chi').\]

\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)

1.       Use position representation        \[ \hat{q} \to q ; \qquad p\to -i\hbar \pp{q} \]      for the operators  \(\hat{p},\hat{q}\)  and prove the following relations    \begin{equation}\label{qm-que-02001;1}      [ \hat{q},\hat{p}^N] = i N\hbar \hat{p}^{N-1}; \qquad\qquad       [\hat{p},\hat{q}^N]  =-i N\hbar \hat{q}^{N-1}.      \end{equation}       Note you could have equally well used the momentum representation  \[ \hat{q} \to  i\hbar \pp{p}\qquad \hat{p} \to p \]   
2.       Prove the above commutators  \eqref{qm-que-02001;1} using canonical commutation relations,  \([q,p]=i\hbar\), only. Do  not use any  representation  for position or momentum operator.

Question: Show that the time period of a relativistic oscillator as  function of amplitude $a$ is given by $$ T\approx \frac{2\pi}{\omega}\left[ 1 + \frac{3\omega^2a^2}{16 c^2} + \cdots \right] $$ where the restoring force is $- kx$ and $k=m \omega2 $.

\(\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\)

A point charge moves with velocity \(\vec{v}\) in presence of electric field \(\vec{E}\) and magentic field \(\vec{B}\). The Lorentz force on the charged particle is
\[\vec{F}= q\Big\{ \vec{E} + \frac{1}{c}\vec{v}\times\vec{B} \Big\}\]
In terms of the scalar potential \(\phi\) and vector  potential \(\vec{A}\) related to the electric and magnetic fields by
\[\vec{E}= -\nabla \phi- \frac{1}{c} \pp[\vec{A}]{t}, \quad
\vec{B}=\nabla\times \vec{A}.\]
the Lorentz force becomes \[\vec{F} = q\Big\{ - \nabla\phi -\frac{1}{c} \pp[\vec{A}]{t} +
\frac{1}{c} \vec{v}\times(\nabla \times \vec{A})\Big\} \]
Determine the generalzed potential for this system and write the Lagrangian.

The interaction of \(\Lambda^0\) hyperon, responsible for decay into a proton
and a \(\pi^-\), is given by \[ H_\text{int} =  \bar{\psi}_p(x) ( g -
g^\prime\gamma_5)\psi_\Lambda(x) \phi_\pi ^\dagger + h.c. \]
(a) Give examples of three virtual processes allowed in the first order of this interaction term.

(b) Show that the partial decay rate of \(\Lambda^0 \longrightarrow p + \pi^-\) is given by
\[ \Gamma = \frac{1}{4\pi}\frac{|\vec{p}|}{M_\Lambda}\left(|g|^2(E_p+M_p) + |g^\prime|^2
(E_p-M_p) \right)\]

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\(\newcommand{\average}[2]{\rangle#1|#2|#1\rangle}\) The state of a quantum system is represented by a two component vector \[ f = \begin{pmatrix}1+3i \\3-i \end{pmatrix}\] For the dynamical variable \(Y\) \[Y = \begin{pmatrix}0 & -i\\i  & 0\end{pmatrix}\] compute the average $\average{f}{Y},\average{f}{Y^2} $  and the uncertainty $\Delta Y$ for the
variable $Y$ where \begin{equation*} Y= \begin{pmatrix} 0 & -i \\ i &0      
\end{pmatrix} \end{equation*} Give your remarks, if any, on the answer you obtained.

  Let
      $$L_x =yp_z-zp_y,\ \  L_y=zp_x-xp_z, \ \ \mbox{ and } L_z=xp_y-yp_x $$  be the angular momentum operators. Prove any one of the following.
          $$ [ L_x, L_y] = i \hbar L_z $$
          $$ [ L_y, L_z] = i \hbar L_x $$
          $$ [ L_z, L_x] = i \hbar L_y $$
      Use the fundamental commutators,      \[[x,p_x]=i\hbar,\quad [y,p_y]=i\hbar,\quad [z,p_z]=i\hbar,\]        and the identities involving commutators.

Assuming \(\frac{g}{4!} \pi^4(x)\) interaction for  neutral pions  of mass \(m\), find

1.    transition probability per unit time per unit volume for \(\pi - \pi\) scattering 
2.   compute the total cross section for the scattering process and show that  \[ \frac{d\sigma}{d\Omega}= \frac{g^2}{64\pi^2 E_\text{cm}^2}\]Be careful about normalization of two identical pion states.