4es04final - Page 1 of 7 Department of Physics University...

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Unformatted text preview: Page 1 of 7 Department of Physics University of California San Diego Modern Physics (4E) Prof. V. Sharma Final Exam (June 11, 2004) Problem 1: Mystery Metal:[20 pts] Photons of wavelength 450nm are incident on a metal. The most energetic electrons ejected from the metal are bent into a circular arc of radius 20cm in a magnetic field whose strength is equal to 2.0!10-5T. What is the work function of the metal? Problem 2: Quantum Pool:[20 pts] An x-ray photon of wavelength 0.02480nm strikes a free stationary electron. The photon scatters off at 90o with respect to the direction of incidence. Determine (a) the momentum of the incident photon (b) the momentum of the scattered photon (c) the kinetic energy of the scattered electron, and (d) the wavelength of the scattered photon. Problem 3: Across The Universe! :[20 pts] An air gun is used to shoot 1.0g particles at 100m/s through a bore (hole) of diameter 2.0mm. How far from the rifle must an observer be to see the beam spread by 1.0cm because of the uncertainty principle? Problem 4: Harmonic Oscillator [30 pts] Consider a 3D harmonic oscillator, described by the potential V (x, y, z) = 1 m! 2 (x 2 + y 2 + z 2 ) . (a) What is the energy of the ground state of 2 this system? What is the degeneracy of this energy? (b) Write down the wavefunction of the ground state. (c) Find the expectation value xy for the ground state. (d) What is the energy of the first excited state? What is the degeneracy of this energy? Problem 5: Step On It: [30 pts] A free particle of mass m with wave number k1 is traveling to the right. At x=0, the potential jumps from zero to V0 and remains at this value for ! 2 k12 = 2V0 , what is the wave number positive x. (a) If the total energy is E = 2m k2 in the region x>0 ? (b) Calculate the transmission ( T ) coefficient at the Page 2 of 7 potential step. (c) If 106 particles with wave number k1 are incident upon the potential step, how many particles are expected to continue along the positive x direction ? (d) How does your answer to part (c) compare with the classical predictions? Problem 6: Dj vu: [30 pts] Consider a particle in a 1-D Harmonic oscillator potential ( U (x) = m! 2 x 2 ) described initially by a wave that is a superposition of the ground state and the first excited states of the oscillator: !(x,0) = C[" 0 (x) + " 1 (x)] (a) show that the value 1 / 2 normalizes this wavefunction assuming "1 and "2 are themselves normalized. (b) Find the expression for !(x,t) at any later time t. (c) Show that the average energy in this state is the arithmetic mean ( E1 + E2 ) / 2 of the ground state and the first excited state energies E1 and E2 . (d) Show that the average particle position oscillates with time as < x >= x0 + Acos(!t) where x0 = 1 2 and ! = ( E2 " E1 ) / ! . 1 2 (" x |! 0 * |2 dx + " x | ! 1 |2 dx and A = " x! 0! 1dx ) Problem 7: Lithium: The Musical! [30 pts] For the ground state of the Li++ ion, calculate the average (a) potential energy and (b) kinetic energy. Problem 8: The Incredible Shrinking Atom:[20 pts] A collection of hydrogen atoms is placed in a 3.5 T magnetic field. Ignoring the effects of spin, what are the possible wavelengths of photons emitted when an atom transitions from a 3s state to a 2p state? Page 3 of 7 Page 4 of 7 Page 5 of 7 Page 6 of 7 Page 7 of 7 ...
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This note was uploaded on 02/12/2012 for the course PHYS 260 taught by Professor Davidhanna during the Spring '10 term at McGill.

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