Chris Parkes 4th Year Particle Physics I Option Question Sheet Section I – Relativistic Quantum Mechanics 1. Starting from the relativistic expression for energy, E2-p2c2=m2c4,make the standard Quantum Mechanical substitutions to derive the Klein-Gordon Equation. 2. Show that the wave equation for a free particle, ( x, t ) N exp[ i ( p.x Et ) / ] satisfies the Klein-Gordon equation, 2 ( x, t ) 2 c 2 2 ( x, t ) m 2 c 4 ( x, t ) and determine the Energy E. 2 t Show that the complex conjugate of the above solution also satisfies the K-G equation and determine the energy in this case. Comment on the energies obtained. 2 3. A beam of positively charged pions is incident on a one-dimensional potential step of height eVo, i.e. V(x)=0 for x<0 V(x)= Vo for x>0 p p' Show that the reflection coefficient,R, for the pions is R p p' 2 where 1 pc ( E 2 m0 c 4 ) 2 1 p' c [( E eVo ) 2 m0 c 4 ] 2 2 Discuss the physical interpretation of the case when eVo > E+moc2 and p’<0 What experimental situation does the choice eVo > E+moc2 and p’>0 represent ? ( x, t ) H ( x, pˆ ) ( x, t ) , where H, the t 3 Hamiltonian, is H ic i mc 2 and p̂ i is equivalent to the xi i 1 Klein-Gordon equation providing i 2 1; 2 1; i i 0; i j j i 0(i j ) 4. Show that the Dirac Equation, i 5. The relativistic wave equation for pions has two solutions. Why does the corresponding equation for electrons have four solutions ? 6. Describe the process e+e- e+e- in terms of hole theory. Section II – Reaction Kinematics 1. Show that in a fixed target experiment with particle a incident on particle b, the square of the centre of mass energy, S, is given by the expression S ma2 2 E a mb mb2 2. In a colliding beam experiment, particles a and b, of same mass and equal but opposite momenta interact to give particles c and d. Show that the centre of momentum of particle c, p c* , is given by the 1 expression p c* p d* {[W 2 (mc md ) 2 ][W 2 (mc md ) 2 ]} 2 / 2W and that the total energy of particle c is given by E c* (W 2 mc2 md2 ) / 2W where W is the total centre of mass energy. What is the corresponding expression for the centre of mass energy of particle d? 3. Show that the invariant mass of a pair of photons of energies, E1 , E2 with angle between their directions is given by the expression m 2 4 E1 E2 sin 2 ( / 2) . Consider the decay of a particle of mass m energy E into 2 photons, and show that the minimum opening angle between the photons is given by sin( / 2) m / E . A particle of energy 10 GeV decays to two photons with opening angle 2. Could this particle be an meson (of mass 0.549 GeV) or a meson (of mass 0.135 GeV)? 4. Define four momentum transfer. In an elastic scattering experiment, particle a is scattered through an angle * in the centre of mass system. Show that the four momentum transfer, q, is given by the expression q 2 2 p *2 (1 cos * ) , where p* is the centre of mass momentum of a. Show that in the limit that the masses of the particles may be neglected, q is proportional to the total centre of mass energy. 5. The experimental cross section for e e in a colliding ring experiment at high energy can be fitted to the equation d 2 ( )(1 cos 2 * ) where d * is an element of solid angle d * 16 E 2 into which the - emerges, * is the angle the - makes with the incident e- and E is the energy of the particles in each beam. Show that 4 2 the total cross section, , is given by where S Ecm and 3S is the fine structure constant. Calculate the total cross section for e e at CM energies of 2 GeV and 40 GeV. What are the corresponding cross- sections for e e ? The masses of the electron, muon and tau lepton are 0.511 MeV, 105.6 MeV and 1777 MeV respectively. 6. In a colliding beam experiment, define luminosity and explain how it is related to the interaction rate. Obtain an expression for luminosity in terms of the design features of an accelerator and discuss how the features can be optimised to improve overall performance. Section III – Feynman Diagrams 1. Draw the topologically distinct QED Feynman diagrams which contribute to the following processes in lowest order: a. e e b. e e e e 2. In lowest order, the process e e is given by the following Feynman diagram. Estimate the distance between the vertices at energies much larger than the masses of the particles in (a) the rest frame of the electron (b) the centre of mass frame. Check the consistency of these estimates by considering the Lorentz contraction in going between the electron rest frame and the centre of mass frame. e- e+ 3. Which of the following annihilation processes are possible: with n=1,2,3,4 e e n For each invalid process, give a reason why this is so. For each valid process, draw at least one Feynman diagram to illustrate it. At a given interaction energy, how would you expect the annihilation rates to compare for the different valid processes ? 4. In electron positron colliders, leptons scatter freely from each other and we do observe free leptons. In high energy proton colliders, quarks also freely scatter from each other but yet we do not observe free quarks. Explain this paradox. 5. Draw Feynman / quark flow diagrams for the following processes and classify them as strong, electromagnetic or weak decays: a. p n b. n p c. 0 e e e e d. D K Section IV – Symmetries, Invariances and Conservation Laws 1. Use the standard commutation relations for angular momentum operators to show that L and S remain good quantum numbers if the spin dependent forces arise from a simple spin-orbit interaction i.e. if ˆ Sˆ H H 0 L where ˆ H , Sˆ 0 H0 ,L 0 and is a constant. 2. The deuteron is a bound state of two nucleons with spin-1 and positive parity. Show that it may only exist in the 3S1 and 3D1 states of the pn system. 3. Electric dipole transitions between atomic states are characterized by the selection rule l 1 . Hence find the parity of the photon. 4. The intensity of the electrons emitted in the decay of polarized cobalt-60 nuclei is found to be consistent with the form: I ( , v) 1 vc cos where v is the magnitude of the electron velocity and theta is the angle between its direction and the cobalt-60 spin. Deduce the value of the coefficient alpha by considering events in which the electron is emitted in the direction of the decaying nuclei. The spins of the cobalt and Nickel nuclei are J=5, J=4 respectively. Orbital angular momentum may be neglected.[Hint: consider the relativistic limit for v and compare with neutrinos] 5. Draw a Feynman diagram to show that D 0 D 0 mixing may occur. 6. Define Helicity and show that it is a pseudo-scalar quantity. 7. Show that the total decay rates for the reactions K 0 e e and K 0 e e are equal if CP is conserved.