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Unformatted text preview: PHY6938 Proficieny Exam Fall 2002 September 13, 2002 Modern Physics and Quantum Mechanics
1. Consider a one-dimensional step potential of the form V (x) = 0 x<0 V0 x 0 where V0 > 0. A particle with total energy E > V0 and mass m is incident on the step potential "from the left" (in other words: the particle starts at negative values of x and travels toward positive values of x). (a) Use the time-independent Schrdinger equation to determine the form o of the particle's wave function in the two regions x < 0 and x 0. (b) Derive expressions for the probabilities that the particle is reflected (R) and transmitted (T ). Hint: Recall that the probability density current is given by j(x) = Re h , im x and that R and T are ratios of probability density currents. 2. The nucleus 113 Cd captures a thermal neutron having negligible kinetic energy, producing 114 Cd in an excited state. The excited state of 114 Cd decays to the ground state by emitting a photon. Find the energy of the photon. 3. A current of quantum-mechanical particles of mass m can be written as J =- i h [ - ] , 2m where is the wave function. Assume that the particles move through a region of space where the potential is complex V = Vr - iVi . Show that particles are being annihilated at a rate 2 R = Vi h per unit volume. Hint: Use the time-dependent Schrdinger equation to obtain "the material o derivative" ( ) + J . t 4. Cosmic ray photons from space are bombarding your laboratory and smashing massive objects to pieces. Your detectors indicate that two fragments, each of mass m0 , depart such a collision moving at speed 0.6 c at angles of 60 relative to the photon's original direction of motion. (a) In terms of m0 and c, what is the energy of the cosmic ray photon ? (b) In terms of m0 , what is the mass M of the particle being struck (assumed to be initially stationary) ? ...
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