Set 3
\[ \newcommand\v[1]{\boldsymbol{#1}} \newcommand\vdot[2]{\v{#1}\cdot\v{#2}} \]
\[ \newcommand{\dd}[2][]{\mathrm{d}^{#1}#2} \]
\[ \newcommand\f[2]{\dfrac{#1}{#2}} \newcommand\da[2]{\dfrac{d #1}{d #2}} \newcommand\daa[2]{\dfrac{d^2 #1}{d #2^2}} \newcommand\pa[2]{\dfrac{\partial #1}{\partial #2}} \newcommand\pal[3]{\left.\dfrac{\partial #1}{\partial #2}\right|_{#3}} \newcommand\paa[2]{\dfrac{\partial^2 #1}{\partial #2^2}} \newcommand\pab[3]{\dfrac{\partial^2 #1}{\partial #2 \partial #3}} \def\half{\frac{1}{2}} \]
\[ \newcommand\ket[1]{\left|#1\right\rangle} % |x> \newcommand\exp[1]{\left\langle#1\right\rangle} % <x> \newcommand\lr[1]{\left(#1\right)} % (x) \newcommand\lrs[1]{\left[#1\right]} % [x] \newcommand\lra[1]{\left|#1\right|} % |x| \newcommand\lrc[1]{\left\{#1\right\}} % {x} \newcommand\lrl[3]{\left.#1\right|_{#2}^{#3}} % x_limits \]
\[ \def\ie{\text{i.e.}\quad} \]
\[ \newcommand{\me}[3][\rightarrow]{{#2}#1{#3}} % Typical reaction \newcommand{\nr}[3]{{#1}\,({#2})\,{#3}} % Nuclear reactions \newcommand{\ele}[4]{_{#2}^{#3}{\textbf{#1}}_{#4}} \newcommand{\el}[2]{^{#2}\text{#1}} \newcommand{\til}[2][\mathbf{\sim}]{\underset{#1}{{#2}}} \newcommand{\b}[1]{\overline{#1}} \]
\[ \newcommand{\coulomb}[1]{\dfrac{#1}{4\pi\epsilon_0}} \def\degree{^*} \]
\[ \def\amu{\text{ amu}} \def\J{\text{ J}} \def\K{\text{ K}} \def\s{\text{ s}} \def\second{\text{ s}} \def\Psecond{\text{ s}^{-1}} \def\m{\text{ m}} \def\mm{\text{ mm}} \def\km{\text{ km}} \def\fm{\text{ fm}} \def\mum{\text{ $\mu$m}} \def\mus{\text{ $\mu$s}} \def\mPERs{\text{ ms$^{-1}$}} \def\eV{\text{ eV}} \def\keV{\text{ keV}} \def\GeV{\text{ GeV}} \def\MeV{\text{ MeV}} \def\MeVc{\text{ MeV/c$^2$}} \def\KeVc{\text{ KeV/c$^2$}} \newcommand\ttp[1]{\times 10^{#1}} % To The Power \]
\[ \def\en{\nu_e} \def\Aen{\b{\nu_e}} \def\mun{\nu_\mu} \def\Amun{\b{\nu}_\mu} \]
\[ \def\uraniumA{\ele{U}{92}{233}{}} \def\uraniumB{\ele{U}{92}{235}{}} \def\uraniumC{\ele{U}{92}{238}{}} \def\carbonB{\ele{C}{6}{14}{}} \def\nitrogenB{\ele{N}{7}{14}{}} \def\oxygenA{\ele{O}{8}{14}{}} \def\oxygenB{\ele{O}{8}{16}{}} \]
\[ % \def\com{\text{\scriptsize CM}} \def\com{\mathrm{\scriptsize CM}} \]
\[ \def\l{l} \]
\[ \def\g{\gamma} \]
Problems
Problem 3.1(Timescales & Energy Widths)
The Energy-Time Heisenberg’s Uncertainty Principle can be expressed as: \[ \Delta E \Delta t \approx \hbar \] Explain what this means?
What is the timescale associated with an energy spread of 150 MeV? (Remember that \(\hbar c=197\MeV\cdot\fm\))
Consider an experiment where you prepare a large number of identical atomic systems in an excited state and allow them to decay. Based on your understanding of quantum mechanics, sketch:
- the energy spectrum of the photons that you would collect and
- the variation of the number of excited atoms with time.
Problem 3.2(Unstable States and Resonances)
We know that a particle characterised by a decay constant \(\lambda\) decays randomly in time, with survival probability \[ p(t) = e^{-\lambda t}. \]
One way to incorporate this exponential decay into a quantum-mechanical description is to introduce an explicit time dependence into an otherwise stationary state of energy \(E_0 = \hbar \omega_0\). \[ \Psi(\vec r, t) = \psi_0(\v r)e^{-i \omega_0 t}\, \sqrt{\lambda}e^{-\lambda t/2}. \tag{1.1}\]
Given that \(\psi_0(\v r)\) is normalised, \[ \int_{\text{all space}} |\psi_0(\v r)|^2\,\dd[3]{r} = 1, \] show that Equation 1.1 implies a consistent normalisation in both space and time via:
\[ \int_\text{All space} \int_{t=0}^{t=\infty} |\Psi(\vec r, t)|^2 \dd[3]{r} \,\dd t = 1 \]
The energy distribution of an unstable state can be obtained from the Fourier transform (Fourier Transforms) of its time dependence.
Starting from the time factor \(e^{-\lambda t/2}\,e^{-i\omega_0 t}\) (for \(t \ge 0\)), show that the associated Fourier transform (viewed as an energy amplitude) has the form \[ A(\omega)\propto \frac{1}{\omega-\omega_0+i\lambda/2}. \tag{1.2}\]Equivalently, in terms of energy show that \[ A(E)\propto \frac{1}{E-E_0+i\Gamma/2}, \tag{1.3}\]
where, \[ \Gamma = \frac{\hbar}{\tau}. \tag{1.4}\]
Use this amplitude to obtain the corresponding energy probability distribution (the Breit–Wigner or Lorentzian form): \[ P(E) \propto \frac{1}{(E-E_0)^2 + (\Gamma/2)^2}. \tag{1.5}\]
Use your preferred plotting software to plot the Breit–Wigner distribution (Equation 1.5) for several values of \(\Gamma\) and two values of \(E_0\).
The parameter \(\Gamma\) is known as the width of the resonance.
Show that \(\Gamma\) is equal to the full width at half maximum (FWHM) of the Breit–Wigner distribution.
Problem 3.3(A Matter of Survival)
Many elementary particles are unstable and decay into lighter particles. These decays occur randomly in time, but with a well-defined statistical regularity. Specifically, the probability that a particle decays in a short time interval \(\dd{t}\) is \(\lambda\,\dd{t}\), where \(\lambda\) is a constant known as the decay constant.
Let \(p(t)\) denote the probability that the particle has not decayed by time \(t\) (that is, it has survived up to time \(t\)).
By considering the change in \(p(t)\) over a short interval \(\dd t\), show that \[ p(t) = e^{-\lambda t}. \tag{1.6}\]
Using the survival probability \(p(t)\) show that the mean (average) lifetime is \[ \langle t \rangle = \frac{1}{\lambda}. \tag{1.7}\]
Now consider an ensemble of \(N_0\) identical, non-interacting particles prepared at \(t = 0\).
Explain why the expected number of particles remaining at time \(t\) is \[ N(t) = N_0\,e^{-\lambda t}. \tag{1.8}\]Demonstrate that the half-life of these particles, \(t_{1/2}\), can be expressed as: \[ t_{1/2} = \frac{\ln 2}{\lambda} \]
If a particle has \(n\) decay channels with associated lifetimes \(\tau_1, \tau_2, \ldots, \tau_n\), show that the overall mean lifetime is related to the individual lifetimes by: \[ \frac{1}{\lambda} = \frac{1}{\sum_{i=1}^{n}\tau_i} \]
Problem 3.4(Pionic Deuterium)
The following reaction shows the subsequent decay into two neutrons of a ‘pionic deuterium’ nucleus, which was formed by the absorption of a pion (\(\pi^-\)) by a deuteron (\(d\)).
\[ \me{\pi^-+d^+}{n+n} \]
Determine the intrinsic parity of the pion based on this reaction and the following information.
- The spins of the pion, deuteron and neutron are \(0\), \(1\) and \(\frac{1}{2}\), respectively.
- The deuteron has even intrinsic parity.
- ‘pionic deuterium’ is typically formed in the \(s\) states.
Please clearly indicate the justification for the steps of your solution.
Problem 3.5(About a Neutron Star)
A neutron star is a compact, dense object made of degenerate neutrons, having a density similar to that in the central part of a heavy nucleus.
If the density of nuclear matter is 0.17 nucleons/fm\(^3\) or \(2.8 \times 10^{17}\) kg/m\(^3\), what is the radius of a neutron star having a mass one and a half times that of the sun? (One solar mass \(= 2.0 \times 10^{30}\) kg.)
A neutron star is one of the possible remnants of a supernova explosion such as SN 1987a, the one which took place in the Large Magellanic Cloud 15,000 light years away and was first observed on Earth on February 24, 1987. When the core of a large star exhausts its nuclear fuel, there is no longer the thermal pressure to counterbalance the gravitational force, and the core of the star collapses. For simplicity, we can consider that all the material in the core of the collapsing star is in the form of \({}^{56}\text{Ni}\) made of 28 neutrons and 28 protons. A nickel atom is electrically neutral with 28 electrons outside. Because of the tremendous gravitational force, the protons in \({}^{56}\text{Ni}\) change into neutrons by capturing the atomic electrons through the reaction \[ p + e^- \rightarrow n + \nu_e \] Calculate the number of neutrinos released in converting 1.5 solar masses of \({}^{56}\text{Ni}\) atoms into neutrons during the gravitational collapse.
- If the total cross section for a neutrino to interact with each nucleon is \(10^{-48} \, \text{m}^2\), how many reactions due to the neutrinos from such a gravitational collapse can one expect in a detector on Earth made of 3000 tons of water? Compare this with the number of events (12) observed with a similar detector at Kamioka due to supernova SN 1987a.
- Assuming that the average energy of each neutrino is 10 MeV, calculate the total amount of energy carried away by the neutrinos from the gravitational collapse. Compare this value with the rest mass energy of the sun.