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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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Classically and nonrelativistically, we say that the energyof a massive free particle is just its kinetic energy. (a) With this assumption, show that the classical particle velocityis. (b) Show that this velocity and that of the matter wave differ by a factor of 2. (c) In reality, a massive object also has internal energy, no matter how slowly it moves, and its total energyis, where.Show thatisand thatisIs there anything wrong with it? (The issue is discussed further in Chapter 6.)
A particle is “thermal” if it is in equilibrium with its surroundings – its average kinetic energy would be. Show that the wavelength of a thermal particle is given by
In Exercise 45, the case is made that the position uncertainty for a typical macroscopic object is generally so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. Here we gain same idea of how small an object would have♦ to be before quantum mechanics might rear its head. The density of aluminum is , is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty ofper decade. How small would it have to be for its position uncertainty to be at least as large asof its radius?
To how small a region must an electron be confined for borderline relativistic speeds sayto become reasonably likely? On the basis of this, would you expect relativistic effects to be prominent for hydrogen's electron, which has an orbit radius near? For a lead atom "inner-shell" electron of orbit radius?
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.)
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