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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.
The proton and electron had been identified by 1920, but the neutron wasn't found until 1932. Meanwhile, the atom was a mystery. Helium, for example, has a mass about four times the proton mass but a charge only twice that of the proton. Of course, we now know that its nucleus is two protons and two neutrons of about the same mass. But before the neutron's discovery, it was suggested that the nucleus contained four protons plus two electrons, accounting for the mass (electrons are "light") and the total charge. Quantum mechanics makes this hypothesis untenable. A confined electron is a standing wave. The fundamental standing wave on a string satisfies , and the "length of the string" in the nucleus is its diameter., so, the electron's wavelength could be no longer than aboutAssuming a typical nuclear radius of determine the kinetic energy of an electron standing wave continued in the nucleus. (Is it moving "slow" or "fast"?) The charge of a typical nucleus is +20e , so the electrostatic potential energy of an electron at its edge would be(it would be slightly lower at the center). To escape. the electron needs enough energy to get far away, where the potential energy is 0. Show that it definitely would escape.
Question: If a particle鈥檚 position uncertainty is zero, what can be said of its momentum uncertainty? If a particle鈥檚 position uncertainty is infinite, what can be said of its momentum uncertainty?
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?
A 1 fs pulse of laser light would be long. What is the range of wavelengths in a 0.3Imlong pulse of (approximately)600nmlaser light?
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