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Chapter 4: Q41E (page 136)
The uncertainty in the position of a baseball of mass.What is the minimum uncertainty in its speed?
Estimate the uncertainty in the speed of the baseball is .
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One of the cornerstones of quantum mechanics is that bound particles cannot be stationary-even at zero absolute temperature! A "bound" particle is one that is confined in some finite region of space. as is an atom in a solid. There is a nonzero lower limit on the kinetic energy of such a particle. Suppose minimum kinetic energy of width . Obtain an approximate formula for its minimum kinetic energy.
Electrons are accelerated through a 20 V potential difference producing a monoenergetic beam. This is directed at a double-slit apparatus of 0.010 mm slit separation. A bank of electron detectors is 10 m beyond the double slit. With slit 1 alone open, 100 electrons per second are detected at all detectors. With slit 2 alone open, 900 electrons per second are detected at all detectors. Now both slits are open.
(a) The first minimum in the electron count occurs at detector X. How far is it from the center of the interference pattern?
(b) How many electrons per second will be detected at the center detector?
(c) How many electrons per second will be detected at detector X?
Question: Analyzing crystal diffraction is intimately tied to the various different geometries in which the atoms can be arranged in three dimensions and upon their differing effectiveness in reflecting waves. To grasp some of the considerations without too much trouble, consider the simple square arrangement of identical atoms shown in the figure. In diagram (a), waves are incident at angle with the crystal face and are detected at the same angle with the atomic plane. In diagram (b), the crystal has been rotated 450 counterclockwise, and waves are now incident upon planes comprising different sets of atoms. If in the orientation of diagram (b), constructive interference is noted only at an angle, at what angle(s) will constructive interference be found in the orientation of diagram (a)? (Note: The spacing between atoms is the same in each diagram.)
Question: Atoms in a crystal form atomic planes at many different angles with respect to the surface. The accompanying figure shows the behaviors of representative incident and scattered waves in the Davisson-Germer experiment. A beam of electrons accelerated through 54 V is directed normally at a nickel surface, and strong reflection is detected only at an angle of 500.Using the Bragg law, show that this implies a spacing D of nickel atoms on the surface in agreement with the known value of 0.22 nm.
In the hydrogen atom, the electron鈥檚 orbit, not necessarily circular, extends to a distance of a about an angstrom from the proton. If it is to move about as a compact classical particle in the region where it is confined, the electron鈥檚 wavelength had better always be much smaller than an angstrom. Here we investigate how large might be the electron鈥檚 wavelength. If orbiting as a particle, its speed at could be no faster than that for circular orbit at that radius. Why? Find the corresponding wavelength and compare it to . Can the atom be treated classically?
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