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Chapter 5: Q69E (page 191)
does the wave function have a well-defined momentum? Explain.
The constant in the front multiplies the initial function and two opposing moving plane waves are added, so the momentum is not clearly defined.
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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?
To determine the two bound state energies for the well.
We learned that to be normalizable, a wave function must not itself diverge and must fall to 0 faster than as x get large. Nevertheless. We find two functions that slightly violate these requirements very useful. Consider the quantum mechanical plane wave Aei(kx-ax)and the weird functionpictured which we here call by its proper name. the Dirac delta function.
(a) Which of the two normalizability requirements is violated by the plane wave, and which by Dirac delta function?
(b) Normalization of the plane wave could be accomplished if it were simply truncated, restricted to the region -b<x<+bbeing identically 0 outside. What would then be the relationship between b and A, and what would happen to A as b approaches infinity?
(c) Rather than an infinitely tall and narrow spike like the Dirac delta function. Consider a function that is 0 everywhere except the narrow region where its value is a constant B. This too could be normalized, What would be the relationship between s and B, and what would happen to B as s approaches 0? (What we get is not exactly the Dirac delta function, but the distinction involves comparing infinities, a dangerous business that we will avoid.)
(d) As we see, the two "exceptional" functions may be viewed as limits of normalizable ones. In those limits, they are also complementary to each other in terms of their position and momentum uncertainties. Without getting into calculations, describe how they are complementary.
A half-infinite well has an infinitely high wall at the origin and one of finite height U0 at x= L . Like the finite well, the number of allowed states is limited. Assume that it has two states, of energy E1 and E2 , where E2 is not much below U0. Make a sketch of the potential energy, then add plausible sketches of the two allowed wave functions on separate horizontal axes whose heights are E1 and E2 .
When is the temporal part of the wave function 0 ? Why is this important?
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