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QM-05 :: RIse of Quantum Theory

This module is about old quantum theory. Problems include application of Bohr Sommerfeld Quantization rule, uncertainty principle, useful ideas and important developments before birth of quantum mechanics.

kapoor kapoor's picture 6 days 4 min ago

Question/QFT-15005


The interaction Hamiltonian for pion decay \(\pi^-\longrightarrow e^- +
\bar{\nu} \) can be written as \[\Hsc_\text{int} = \frac{g}{\sqrt{2}}\bar{\psi}_e(x)\gamma_\mu(1-\gamma_5)\psi(x)_\nu \partial^\mu \phi_\pi^-(x) + h.c. \]

  1. Show that the decay rate is given by \[\Gamma = \frac{g^2}{8\pi} \frac{m_e^2(m_\pi^2-m_e^2)^2}{m_\pi^3}.\]
  2. Assuming that the coupling constant for \(\pi^-\longrightarrow \mu^- +\bar{\nu} \) is equal to that for the electron decay, calculate the branching ratio \[\frac{\Gamma(\pi^-\longrightarrow \mu^- +\bar{\nu})} {\Gamma((\pi^-\longrightarrow e^- + \bar{\nu})} \]
kapoor kapoor's picture 6 days 7 hours ago

QFT/15004

{The original four fermion interaction for beta decay of  neutron
 \[n \longrightarrow p + e^- + \bar{\nu} \]
 is of the of form
 \[ \bar{\psi}_p(x)\gamma_\mu\psi_n(x) \bar{\psi}_\nu(x) \gamma^\mu \psi_e(x) +  h.c.\]
Now consider other processes given below. Which of these processes (real or virtual) are permitted and which ones are not permitted by the above interaction in the first order?

  1.   \( \bar{p} \longrightarrow \bar{n} + e^-  +\bar{\nu} \);
  2.   \( \bar{p} \longrightarrow \bar{n} + e^-  +\nu \);
  3.   \( n \longrightarrow p + e^+  + \nu \);
  4.   \( p \longrightarrow n + e^+  + \bar{\nu} \);
  5.   \( \bar{n} \longrightarrow \bar{p} + e^+  + \nu \);
  6.   \( \bar{n} \longrightarrow \bar{p} + e^+  + \bar{\nu} \).


Give brief reason in each case.

kapoor kapoor's picture 6 days 7 hours ago

QFT-15005


The interaction Hamiltonian for pion decay \(\pi^-\longrightarrow e^- + \bar{\nu} \) can be written as \[H_\text{int} = \frac{g}{\sqrt{2}}\bar{\psi}_e(x)\gamma_\mu(1-\gamma_5)\psi_\nu(x) \partial^\mu \phi_{\pi^-}(x) + h.c. \]

  1. Show that the decay rate is given by \[\Gamma = \frac{g^2}{8\pi} \frac{m_e^2(m_\pi^2-m_e^2)^2}{m_\pi^3}.\]
  2. Assuming that the coupling constant for \(\pi^-\longrightarrow \mu^- + \bar{\nu} \) is equal to that for the electron decay, calculate the branching ratio \[\frac{\Gamma(\pi^-\longrightarrow \mu^- + \bar{\nu})}{\Gamma((\pi^-\longrightarrow e^- + \bar{\nu})} \]
kapoor kapoor's picture 6 days 7 hours ago

QFT-14001

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

kapoor kapoor's picture 6 days 7 hours ago

QFT-06003

\(\newcommand{\ket}[1]{|#1\rangle} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\)
Compute the matrix element  \[ \matrixelement{s,\vec{q}}{J_\mu(x)}{r,\vec{p}}\] where \(J_\mu(x)\) is the current for a Dirac particle and \( \ket{s,\vec{q}}\) and \(\ket{r,\vec{p}} \) denote  one particle states of spin half particle with spin and momenta as specified.

kapoor kapoor's picture 6 days 7 hours ago

QFT-05005

\(\newcommand{\ket}[1]{|#1\rangle}
\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\)

For a real free Klein Gordon field, mass \(m\),  compute  \[ \matrixelement{0}{\phi(x)\phi(y)}{\vec{k}, \vec{q}}\] and show that the result is  properly symmetrized wave function for two identical bosons with momenta \(\vec{q},\vec{p}\). Here \(\ket{\vec{k}, \vec{q}}\) is the state with two bosons with momenta \(\vec{k}, \vec{q}\).

kapoor kapoor's picture 6 days 7 hours ago

QFT-05004

 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.

kapoor kapoor's picture 6 days 7 hours ago

QFT-05003

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 \( \int d^3x \big[f_q(x) (f_q(x) \{\partial_0 f_p(x)\} - \{\partial _0 f_q(x)\} f_p(x))\big]\)

                    

kapoor kapoor's picture 6 days 7 hours ago

QFT-05002

Starting from the Lagrangian for a complex scalar field obtain the
Hamiltonian for a free complex Klein Gordon field.
and show that quantized field obeys \(\begin{equation}\big[H, \phi(x)\big] = - i \big( \nabla^2- \mu^2\big) \phi^*(x) \end{equation}.\)

kapoor kapoor's picture 6 days 7 hours ago

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