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Chapter 5: Q24E (page 187)
An electron in the n=4 state of a 5 nm wide infinite well makes a transition to the ground state, giving off energy in the form of photon. What is the photon’s wavelength?
The wavelength of the photon is .
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Equation (5 - 16) gives infinite well energies. Because equation (5 - 22) cannot be solved in closed form, there is no similar compact formula for finite well energies. Still many conclusions can be drawn without one. Argue on largely qualitative grounds that if the walls of a finite well are moved close together but not changed in height, then the well must progressively hold fewer bound states. (Make a clear connection between the width of the well and the height of the walls.)
To determine the classical expectation value of the position of a particle in a box is , the expectation value of the square of the position of a particle in a box isrole="math" localid="1658324625272" , and the uncertainty in the position of a particle in a box is .
Quantum-mechanical stationary states are of the general form . For the basic plane wave (Chapter 4), this is , and for a particle in a box it is . Although both are sinusoidal, we claim that the plane wave alone is the prototype function whose momentum is pure-a well-defined value in one direction. Reinforcing the claim is the fact that the plane wave alone lacks features that we expect to see only when, effectively, waves are moving in both directions. What features are these, and, considering the probability densities, are they indeed present for a particle in a box and absent for a plane wave?
does the wave function have a well-defined momentum? Explain.
Under what circumstance does the integral diverge? Use this to argue that a physically acceptable wave function must fall to 0 faster than does as gets large.
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