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Chapter 4: Q18E (page 134)
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
As a result, were able to demonstrate that :
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The diagram shows the Fourier transform of a Gaussian wave functionthat represents a reasonably well-localized particle.
(a) Determine approximate quantitative values for the wave function's wavelength and for the particle's position uncertainty.
(b) Can you determine the particle's approximate position? Why of why not?
What is the density of a solid'? Although the mass densities of solids vary greatly, the number densities (in ) vary surprisingly little. The value. of course, hinges on the separation between the atoms-but where does a theoretical prediction start? The electron in hydrogen must not as a particle orbiting at a strict radius, as in the Bohr atom. but as a diffuse orbiting wave. Given a diffuse probability, identically repeated experiments dedicated to "finding" the electron would obtain a range of values - with a mean and standard deviation (uncertainty). Nevertheless, the allowed radii predicted by the Bohr model are very close to the true mean values.
(a) Assuming that atoms are packed into a solid typically jBohr radii apart, what would be the number of moles per cubic meter?
(b) Compare this with the typical mole density in a solid of. What would be the value of j?
Question: An electron beam strikes a barrier with a single narrow slit, and the electron flux number of electrons per unit time per unit area detected at the very center of the resulting intensity pattern is . Next, two more identical slits are opened, equidistant on either side of the first and equally “illuminated” by the beam. What will be the flux at the very center now? Does your answer imply that more than three times as many electrons pass through three slits than through one? Why or why not?
The average kinetic energy of a particle at temperatureis. (a) What is the wavelength of a room-temperature ()electron? (b) Of a room-temperature proton? (c) In what circumstances should each behave as a wave?
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.
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