PHYS08045 Special Relativity
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Section A: Special Relativity
A.1 Cosmic rays produce muons in Earth’s upper atmosphere. An experiment is set up to
measure the rate of muons at various altitudes, selecting only those muons with (total) energies of 1 GeV. What fraction of the muon flux at 1000 m altitude would you expect to survive at an altitude of 500 m? (The muon has a rest mass of 106 MeV/c2 and a
characteristic mean lifetime of 2.20µs.)
A.2 In 2100, a space ship is sent from Earth to Epsilon Eridini, 10.5 light years away, travelling
at a constant speed of 0.6c. Immediately upon arrival the space ship sends a radio signal back to Earth.
a) For those who travelled on the space ship, what is the date when the signal is sent?
b) For those on the Earth, what is the date when the signal is received?
A.3 You are a member of the Interstellar Police and have been tasked with determining who fired the first shot in a recent interplanetary war.
The evidence is provided by Luke, a reliable witness located on a ship that was approaching the Centauri system. The system consists of two planets, Alpha and Beta, separated by a constant distance of 200,000 km with respect to one another. Luke was approaching the system at a speed of 0.98c, on a linear path that would have first gone closely past Alpha and then past Beta. While a long way off, he saw a burst of energy first from Beta and then 1.2 seconds later from Alpha.
The Alphans have cited this evidence as clear proof that the Betans fired a powerful laser weapon at them. The Betans strongly refute the claim.
Who will you judge as guilty?
a) Consider Luke as being fixed in an inertial frame S. Alpha and Beta are fixed in frame S′ which is moving with respect to frame S. The relative speed of the two frames is v = 0.98c. In the context of the Lorentz transformations, is v positive or negative?
b) Write down the Lorentz transformations for intervals.
c) Noting the use of ‘primes’, clearly identify which two of ∆x, ∆t, ∆x′ and ∆t′ are given directly in the question.
d) Calculate the appropriate Lorentz factor of the system
e) Apply the Lorentz transformations for intervals to derive values for the two unknown parameters.
f) With the above calculations completed you should now be able to make a judgement. Are the Betans guilty? Explain your answer
Section B: Quantum Mechanics
B.1 Evaluate the commutator 
where 
is the operator for position and 
is the operator
for kinetic energy. Comment briefly on the significance of your result.
B.2 A quantum mechanical particle is confined to a one-dimensional infinite potential well
described by the function V (x) = 0 in the region 0 < x < L, V (x) = ∞ elsewhere. The normalised eigenfunctions for a particle moving in this potential are:
a) Write down the expression for the corresponding probability density function. Sketch the shape of this function for a particle in the ground state (n = 1).
b) Annotate your sketch to show the probability density function for a classical particle moving at constant speed in the well. Give a short justification for the shape of your sketch.
c) Briefly describe, with the aid of a sketch or otherwise, the way in which the quantum and the classical probability density functions are consistent with the correspondence principle for large values of n.
B.3 Consider a potential described by the function:
A quantum mechanical particle having energy E greater than 2V0 is incident from the left (x < 0).
a) Write down, with a short justification, a plausible form for the eigenfunctions in each of the regions defined by the potential.
b) State the boundary conditions that the eigenfunctions must satisfy at the boundaries between the regions; and the equations for the coefficients of the eigenfunctions that result from these conditions. (You do not need to determine these coefficients).
c) Describe qualitatively, with a short justification, the relative wavelengths of the eigenfunctions in each of the regions.
d) Repeat part (a) for the case where the energy E of the incident particle is in the range V0 < E < 2V0 .
2021-12-09