Introduction to Modern Physics Problems from previous Exams 3

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1 Introduction to Modern Physics Problems from previous Exams An electron of mass kg moves along the x axis at a velocity.9c. a. Calculate the rest energy of the electron. b. Calculate its kinetic energy non-relativistically. c. Calculate its kinetic energy relativistically The three primary terms which determine the binding energy of a nucleus are volume, surface and Coulomb, E V, E S, E C energies. a. What is the R and Z dependence of each, where R is the nuclear radius and Z the atomic number. Also indicate the sign of each. i E V ii E S iii E C b. What is the A and Z dependence of each, where A is the number of nucleons. Also indicate the sign of each. i E V / A ii E S / A iii E C / A c. In the space provided in the graph below draw the magnitude of each, as well as the sum of each. Be sure to clearly fill in the energy scale in the vertical axis and the number of nucleons in the horizontal axis at the position of the tic marks. E B A 2006 A a. The major source of energy production in the sun is the proton-proton cycle. Trace the steps of the p-p cycle as we discussed in class. b. If the final result is the fusion of 4 protons into 4 He, calculate the total energy released in the cycle.

2 2006 Draw a graph for the shape of the nucleon-nucleon attractive potential energy, indicating the approximate range and depth Approximate a nucleus consisting of free nucleons in a spherical rigid wall potential with radius R=4 fm. For the isotope 17 O: a. What are the quantum numbers of each of the neutrons and protons? b. What are the energies of each of the neutrons and protons in the isotope 17 O? 2006 a U captures a neutron, followed by asymmetric fission into 2 unbound neutrons and Sr and Xe. Obtain the difference in the binding energy between the initial U and the final Sr and Xe nuclides, and therefore the energy released. b. Calculate the kinetic energy due to the electrostatic repulsion between the Sr and Xe when they are still touching, and show that it is the same order as your answer in part a. above. (note:r = r 0 A 1/3 with r fm.) 2008 a. Write the wave function for a free particle moving in 3-dimensional Cartesian coordinates. b. The relativistic version of the Schroedinger equation is called the Klein-Gordon equation. Using E 2 = p 2 c 2 + m 2 c 4, construct the Klein-Gordon equation by expressing the energy and momentum in terms of differential operators. c. Show that the wave function in part a. is a solution to the Klein-Gordon wave function that was constructed in part c Consider an electron which moves freely in a 2 dimensional infinite square well of side a. a. Write the Schroedinger equation for this case. c. What are the allowed energy levels? d. If a = 10 Angstroms, what is the lowest energy. e. Write the wave function for this state Draw a Feynman diagram for each of the following processes, and identify the exchanged quantum: a. e - + µ + e - + µ + via the electromagnetic interaction. b. e - +µ + e + + µ - via the weak interaction. c. u + u s + s via the strong interaction a. Show that the ordinary non-relativistic Schroedinger equation is not invariant with respect to a simple local phase transformation e iα(x,t). b. State in a couple of sentences what must be done to make it invariant.

3 3. Fill in the table below: 8 3Li β + ν 16 8O stable Fe Sr Pd Pd 2007 In the blank spaces provided in the table, fill in the properties of the particle shown, as well as the energy scales and quark makeup where appropriate. a. What are the approximate masses of the following particles in units of ev? Particle mc 2 Particle mc 2 Proton Electron Neutrino Photon Gluon Weak Boson b. What are the quark contents of the following particles based on their listed properties? Baryon q s Quark content Meson q s Quark content Proton +1 0 p neutron 0 0 p S K

4 2006 From the information on spin, baryon number and strangeness given in the table below, fill in the quark flavor content and decay interaction of each of the following hadrons. Decay interact we

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6 α ( r) α e (r µ ) 2 r 2 1 α e(r µ ) 3π ln r µ r ~r µ = 1 fm α e = ~ π α S ( 33 2n f )ln r Λ 2 r 2 r Λ λ Λ ~ 6 fm.