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Chapter 40: Q. 19 (page 1175)
An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a quantum jump. What is the spring constant of the potential well?
Spring constant of potential well =
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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.
An electron in a rigid box absorbs light. The longest wavelength in the absorption spectrum is. How long is the box?
a. What is the probability that an electron will tunnel through aair gap from a metal to a STM probe if the work function is ?
b. The probe passes over an atom that is鈥渢all.鈥� By what factor does the tunneling current increase?
c. If a current change is reliably detectable, what is the smallest height change the STM can detect?
Model an atom as an electron in a rigid box of length , roughly twice the Bohr radius.
a. What are the four lowest energy levels of the electron?
b. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label to indicate the transition.
c. Are these wavelengths in the infrared, visible, or ultraviolet portion of the spectrum?
d. The stationary states of the Bohr hydrogen atom have negative energies. The stationary states of this model of the atom have positive energies. Is this a physically significant difference? Explain.
e. Compare this model of an atom to the Bohr hydrogen atom. In what ways are the two models similar? Other than the signs of the energy levels, in what ways are they different?
FIGURE Qshows two possible wave functions for an electron in a linear triatomic molecule. Which of these is a bonding orbital and which is an antibonding orbital? Explain how you can distinguish them.
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