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Chapter 4: Q15E (page 134)
How slow would an electron have to be traveling for its wavelength to be at least?
The electron speed is .
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Equation (4-21) expresses a function as a sum of plane waves, each with a coefficient . Equation (4-22) finds the coefficients from the given function . The equations aren't independent statements; in fact, one is the inverse of the other. Equation (4-22) giveswhen is known, and (4-21) does the reverse. Example 4.7calculates from a specific. Show that when this is inserted into (4-21) , the original is returned. Use the Euler formula and the symmetry properties of odd and even functions to simplify your work.
The setup depicted in Figureis used in a diffraction experiment using X-rays ofwavelength. Constructive interference is noticed at angles ofand,but none between. What is the spacingof atomic planes?
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 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?
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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