Set 2

TipInstructions
  • Here are some additional practice questions and activities for you to work on independently.
  • Some of these questions are from previous exams and tests.
  • I will not provide written solutions, as I want to avoid them being circulated to future batches, which would render the practice less effective. However, I am more than happy to discuss and assist you as you work through them.
    Please feel free to email me at (chammika[at]nus.edu.sg) to organise a meeting.

Problems

Problem 2.1(The Fickle Neutron)

We know that the neutron left to its own devices will decay into a proton via \(\beta\)-decay in about 15 minutes via:

\[ \me{n}{p^+ + e^- + \Aen} \]

However, the neutron does not decay when safely tucked away in a nucleus. Let’s try to understand why.

Consider the isotope of hydrogen called heavy-hydrogen or deuterium (\(\ele{H}{1}{2}{}\)). The nucleus of this atom is called a deuteron and is stable. The deuteron is simple and consists of a proton and a neutron. Just by considering the masses of the neutron, proton, electron and deuteron argue that the neutron in a deuteron cannot decay.

Problem 2.2(Reactions & Feynman Diagrams)

  1. \(\quad\me{\mu^-}{e^++\gamma}\)
  2. \(\quad\me{\mun+p}{\mu^+ + n}\)
  3. \(\quad\me{\pi^+}{\pi^0 + e^+ + \en}\)
  4. \(\quad\me{K^-}{\pi^++\pi^-}\)
  5. \(\quad\me{\pi^-+p}{\Xi^- + K^+ + K^0}\)

For each of the reactions shown above:

  • Establish whether the reactions is allowed as shown.
  • If it is not allowed, fiddle with it to get it to work,
  • Draw a suitable Feynman diagram to represent the reaction.

Problem 2.3(Uncertainty Principle and Energy Estimation)

Use the uncertainty relation

\[ \Delta x\,\Delta p \approx \frac{\hbar}{2} \]

to estimate the minimum kinetic energy of an electron confined to a region of width equal to a nucleus \((10^{-15}\ \mathrm{m})\).

Use the relativistic energy–momentum relation and report your answer in MeV. 98.0 MeV

Remember that \(\hbar c=197\MeV\cdot\fm\).

Problem 2.4(Mass of the Yukawa Pion)

  1. In the 1930s, the Yukawa Pion was initially considered to be the particle responsible for mediating the strong interaction. By applying the energy-time formulation of the Heisenberg uncertainty principle (\(\Delta E \Delta t = \hbar/2\)) and visualising the pion moving the width of a nuclei (\(10^{-15}\) m) at the speed of light, arrive at an estimate for the mass of the Yukawa Pion.
    Express your answer in MeV/c\(^2\) 98.7 MeV/c2

  2. You can figure out the ‘mass’ of the particle mediating the electromagnetic interaction by applying the same logic used above to the atom (\(10^{-10}\) m). Do this for good measure, so that you know how reliable the method is.

Problem 2.5(Maximum Kinetic Energy in Particle Decays)

A free neutron and a free muon, each initially at rest in the laboratory, undergo the decays

\[ n \rightarrow p + e^- + \bar{\nu}_e \]

and

\[ \mu^- \rightarrow e^- + \nu_\mu + \bar{\nu}_e. \]

Treat the neutrinos as massless.

  1. Using conservation of energy and momentum, determine the maximum kinetic energy of the electron in each decay.
  2. For each decay, briefly describe the momenta of the other decay products when the electron has its maximum kinetic energy.

Problem 2.6(A Useful Approximation for Neutron Decay)

Physicists often make carefully chosen approximations to simplify a problem while retaining its essential physics.

Reconsider the neutron decay

\[ n \rightarrow p + e^- + \bar{\nu}_e. \]

Because the proton is much more massive than the electron, assume that its recoil kinetic energy is negligible.

  1. Use energy conservation to estimate the maximum kinetic energy of the electron.
  2. Compare your estimate with the result obtained previously.
  3. Comment briefly on whether neglecting the proton’s recoil energy is a useful approximation.