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Chapter 40: Q. 9 (page 1175)
A finite potential well has depth. What is the penetration distance for an electron with energy
(a)
(b) and
(c)?
(a) The penetration distance is .
(b) The penetration distance is .
(c) The penetration distance is .
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A neutron is confined in a -diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than from the edge of the nucleus?
Figure 40.27a modeled a hydrogen atom as a finite potential well with rectangular edges. A more realistic model of a hydrogen atom, although still a one-dimensional model, would be the electron + proton electrostatic potential energy in one dimension:
a. Draw a graph of U(x) versus x. Center your graph at .
b. Despite the divergence at , the Schr枚dinger equation can be solved to find energy levels and wave functions for the electron in this potential. Draw a horizontal line across your graph of part a about one-third of the way from the bottom to the top. Label this line , then, on this line, sketch a plausible graph of the wave function.
c. Redraw your graph of part a and add a horizontal line about two-thirds of the way from the bottom to the top. Label this line , then, on this line, sketch a plausible graph of the wave function.
Consider a particle in a rigid box of length L. For each of the states and :
a. Sketch graphs of . Label the points and .
b. Where, in terms of L, are the positions at which the particle is most likely to be found?
c. Where, in terms of L, are the positions at which the particle is least likely to be found?
d. Determine, by examining your graphs, if the probability of finding the particle in the left one-third of the box is less than, equal to, or greater than . Explain your reasoning.
e. Calculate the probability that the particle will be found in the left one-third of the box
a. Derive an expression for , the wavelength of light emitted by a particle in a rigid box during a quantum jump from
b. In what length rigid box will an electron undergoing a transition emit light with a wavelength of ? This is the wavelength of a ruby laser
Showed that a typical nuclear radius is As you鈥檒l learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well is. To find out how 鈥渇uzzy鈥� the edge of the nucleus is, what is the neutron鈥檚 penetration distance into the classically forbidden region as a fraction of the nuclear radius?
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