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Chapter 4: Q17E (page 134)
Determine the Compton wavelength of the electron, defined to be the wavelength it would have if its momentum were.
Compton Wavelength of the electron
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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
The energy of a particle of massbound by an unusual spring is.
(a) Classically. it can have zero energy. Quantum mechanically, however, though both andare "on average" zero, its energy cannot be zero. Why?
(b) Roughly speaking.is a typical value of the particle's position. Making a reasonable assumption about a typical value of its momentum, find the particle's minimum possible energy.
Roughly speaking for what range of wavelengths would we need to treat an electron relativistically, and what would be the corresponding range of accelerating potentials? Explain your assumptions.
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.
Determine the momenta that can never be measured when a particle has a wave function.
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