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Physics 249 Final Exam Dec 21st 2012 Constants Proton mass: 1.6726x10-27 kg or 938.27 MeV/c2 Electron mass: 9.1094x10-31 kg or 0.51100 MeV/c2 Electron charge: e=1.60x10-19 C ke = 8.99x109 Nm2/C2 𝑘𝑒 𝑒 2 = 1.44𝑒𝑉𝑛𝑚 ℏ = 1.0546x10-34 J*s, 6.5821x10-16 eV*s c = 3.00x108 m/s ℎ𝑐 = 1240𝑒𝑉𝑛𝑚 ℏc = 197.3eVnm The mass energy of 1 atomic mass unit, u is 931.5 MeV Integrals 1) a) What is the minimum energy(in eV) and wavelength(in nm) of a photon needed to ionize the n=3 orbit of He+. b) If you use a photon of wavelength 100nm to ionize the electron in this system what is the stopping potential (in V) for the ionized electron. (15 points) 2) A gold foil is used in a Rutherford experiment to scatter alpha particles with energies 10 MeV. What fraction of the particles are backscattered to angles within 5 degrees of the incoming direction. For the foil. Z=79, thickness = 1um, n = 5.90e28 atoms/m 3 (15 points) 3) a) For the Harmonic oscillator wave functions n=0, 1, 2 prove for all cases the equation. +∞ ∫ 𝜓𝑛∗ 𝑥𝜓𝑚 = 0 −∞ unless 𝑛 = 𝑚 ± 1 b) Interpret what this result means for transitions between different energy states of the harmonic oscillator 𝜓0 = 𝐴0 𝑒 −𝑦 2 /2 𝜓1 = 𝐴1 𝑦𝑒 −𝑦 2 /2 𝜓2 = 𝐴2 (2𝑦 2 − 1)𝑒 −𝑦 𝑦 = √𝛼𝑥, (25 points) 𝛼= ℏ𝜔 2 2 /2 4) Use the Heisenberg uncertainty principle to estimate the size of the hydrogen atom. Calculate the expectation value of r to determine the average radius of the ground state and compare this to the estimate from the uncertainty principle. 1 2 −𝑟/𝑎 0 𝑒 4𝜋 √𝑎03 (20 points) 𝜓=√ 19 5) Consider 10 𝑁𝑒 and 199𝐹 a) What orbitals are filled for the protons and neutrons in these two nuclei? List how many neutrons or protons are in each energy level. Use the notation that lists energy levels by primary quantum number, orbital shell type and total angular momentum, rather than shell number. (Hint – all of the levels that are filled have n=1) b) Which is the unstable isotope and why. c) Write down the decay mechanism that converts the unstable to the stable isotope. d) Calculate the nuclear radius for these isotopes given For these isotopes ∆𝑈 = (19.001880 − 18.998404)𝑢 ∗ ( 931.5𝑀𝑒𝑉 𝑢 ) (15 points) 6) Annihilating electrons and positrons to photons followed by pair production has been one of High Energy physics basic tools for producing and studying new particles. Write Feynman diagram for the following process a) One diagram for 𝑒 + 𝑒 − → 𝜇 + 𝜇 − In some cases more diagrams are possible and you have to add the amplitudes of both interactions. b) Write two diagrams for 𝑒 + 𝑒 − → 𝑒 + 𝑒 − c) Give one example of a particle antiparticle pair that cannot be produced this way. (10 points)