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Consider a ux of particles with wavelength passing through a slit of width d

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Chemistry 120A Problem Set 2;(due September 19, 2014);1. Consider a ux of particles with wavelength passing through a slit of width d and;forming a diraction pattern on a screen as indicated in the schematic Figure 1.;(a) For relatively small, and for a screen a large distance away compared to the;slit size, d, show that sin() /d.;(b) In class, we discussed how spatially conning a plane wave by the slit d leads to;a distribution of directions in which it travels. Consider a particle beam, with;a wavelength given by the deBroglie relation = h/p. Derive a relationship;between the spatial connement of that particle (slit d) and the momentum;distribution that particle takes on after passing through the slit.;(c) Consider a particle of 1 gram travelling at 1 m/s. Find its wavelength and;discuss why one would never be able to observe its diraction pattern.;2. With a neutron pile, neutrons with a wide spread of kinetic energy are let into a;very long block of graphite, as illustrated schematically in Figure 2. Some neutrons;scatter out the side and some continue on in the forward direction. The intensity;of the forward beam, I as a function of neutron wavelength,, looks as shown in;Figure 2. Why is there a threshold, i.e., why is min > 0? What property of;graphite determines the value of min?;3. A particle of mass m moves in an attractive potential V(r) = - exp(-r/a) / r where;and a are both positive. Reduce the equations of motion to an equivalent onedimensional problem, so that you have an eective Vl (r) like we did in class for;the normal coulomb potential. Use the eective potential to discuss the qualitative;nature of the orbits for dierent values of energy (E) and angular momentum (l).;Use at least four dierent values of angular momentum. [Note: This potential could;describe an electron orbiting an atom in a screened coulomb potential where all;the other electrons decrease the attractive potential of the positively charged nucleus;exponentially with distance from the center of the atom.];4. Investigate the Bohr quantization condition for massive objects under the inuence;of a central potential. You attach a mass m= 0.1 kg to an ideal spring of spring;constant k=0.1 kg m/s2. You spin the mass over your head in a horizontal circle of;radius r= 0.5 m.;(a) Use the Bohr quanization condition to determine the energy levels of the circular orbits. What is the quantum number of the 0.5 m orbit?;(b) What is the minimum step outward in radius allowed by Bohr?;(c) What does this suggest for quantum numbers and discrete energy levels in;macroscopic objects?;1;d;Figure 1: Diraction through a slit.;I;neutrons;graphite;I;min;Figure 2: A beam of neutrons passing through a graphite rod. The resulting intensity in;the forward direction is I.;2

 

Paper#16177 | Written in 18-Jul-2015

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