BPHET-141 (ELEMENTS OF MODERN PHYSICS)

(Valid from 1st January, 2023 to 31 st December, 2023)

PART A

1. a) A muon produced in the earth’s atmosphere is travelling with a speed of \(0.80 \mathrm{c}\). As measured in the muon’s frame of reference it has a lifetime of \(1.6 \mu \mathrm{s}\). What is it’s lifetime as measured by an observer on earth.

b) A space ship is measured to be \(180 \mathrm{~m}\) long on the ground. When in flight, its length is measured as \(150 \mathrm{~m}\) by an observer on the ground. What is its speed?

c) Two \(\beta\)-particles move in opposite directions with velocities of \(0.7 c\) in the laboratory frame. Calculate the velocity of one \(\beta\)-particle in the moving frame attached to the other \(\beta\)-particle.

d) A light source emits light of wavelength \(550 \mathrm{~nm}\) while at rest. When the source is moving, the Doppler shifted wavelength of the emitted light is \(430 \mathrm{~nm}\). Is the source of light approaching us or receding from us? Also calculate its speed.

e) Show that when the kinetic energy of a relativistic particle is equal to its rest energy, the speed of the particle is around \(0.866 \mathrm{c}\).

2. a) The stopping potential observed in a certain experiment on photoelectric effect is \(8.0 \mathrm{~V}\). Calculate the maximum kinetic energy and maximum speed of the photoelectrons.

b) Calculate the de Broglie wavelength of an electron in the first Bohr orbit of the hydrogen atom.

c) Estimate the minimum kinetic energy a neutron confined to a nucleus of diameter \(4 \times 10^{-15} \mathrm{~m}\) may have.

d) Show that the wavefunction \(\psi(x)=N \sin k x+i N \cos k x\) is an eigenfunction of the momentum operator.

e) In a region of space a particle has a wavefunction
\[
\psi(x)=\left\{\begin{array}{cc}
A \cos \left(\frac{2 \pi x}{L}\right) & \text { for }-\frac{L}{4} \leq x \leq \frac{L}{4} \\
0 & \text { elsewhere }
\end{array}\right.
\]
Determine the normalization constant \(A\).

PART B

3. a) A particle is in a one dimensional box of length \(a\). If the particle is in the ground state, obtain \(\Delta x\), where \((\Delta x)^2=\left\langle x^2\right\rangle-\langle x\rangle^2\).

b) A particle encounters a step potential of height \(V_0\). Calculate the reflection and transmission coefficient if \(E=1.5 V_0\) ?

c) Calculate the transmission coefficient for an electron of energy \(2.0 \mathrm{eV}\) incident on a potential barrier of \(2.5 \mathrm{eV}\), if the width of the barrier is \(0.50 \mathrm{~nm}\).

d) For a symmetric potential function show that the parity operator commutes with the Hamiltonian.

4. a) Determine B.E. per nucleon for \({ }_{28}^{68} \mathrm{Ni}\) given that the mass of \(\mathrm{Ni}\) is \(63.9280 \mathrm{u}\), \(m_p=1.007825 \mathrm{u}, m_n=1.008665 \mathrm{u}\) and \(m_e=.00054857 \mathrm{u}\).

b) The half-life of an element \(C^{14}\) is 5370 yrs. Detrmine after how much time \(40 \%\) of this sample would have decayed?

c) Show that for light nuclei, the fact that \(Z \cong N\) is explained by the semi empirical formula.

d) Calculate the activity of \(1 \mathrm{~g}\) sample of \(\mathrm{Sr}-90\) whose half life is 28 years. Express your answer in units of Curie \((\mathrm{Ci})\). Take \(1 \mathrm{Ci}=3.7 \times 10^{10}\) disintegrations \(\mathrm{s}^{-1}\).

e) Calculate \(Q\) value of the following nuclear reaction.
\[
{ }_{13} \mathrm{Al}^{27}+{ }_2 \mathrm{He}^4 \rightarrow{ }_{14} \mathrm{Si}^{30}+{ }_1 \mathrm{H}^1
\]
Take \(m\left({ }_{13} \mathrm{Al}^{27}\right)=26.9815 \mathrm{u}, m\left({ }_{14} \mathrm{Si}^{30}\right)=29.9738 \mathrm{u}, m\left({ }_2 \mathrm{He}^4\right)=4.0026 \mathrm{u}\) and \(m\left({ }_1 \mathrm{H}^1\right)=1.0078 \mathrm{u}\).