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Chapter 6: Q1CQ (page 223)
Could the situation depicted in the following diagram represent a particle in a bound state? Explain.
The answer is no.
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Show that if you attempt to detect a particle while tunneling, your experiment must render its kinetic energy so uncertain that it might well be "over the top."
Consider a potential barrier of height . (a) Find a width aroundfor which there will be no reflection of electrons incident upon the barrier. (b) What would be the reflection probability for electrons incident upon the same barrier? (Note: This corresponds to a difference in speed of less than
For a general wave pulse neither E nor p (i.eneither nor k)areweIldefined. But they have approximate values . Although it comprisemany plane waves, the general pulse has an overall phase velocity corresponding to these values.
If the pulse describes a large massive particle. The uncertaintiesarereasonably small, and the particle may be said to have energy and momentum .Usingtherelativisticallycorrect expressions for energy and momentum. Show that the overall phase velocity is greater thancand given by
Note that the phase velocity is greatest for particles whose speed is least.
Given the same particle energy and barrier height and width, which would tunnel more readily: a proton or an electron? Is this consistent with the usual rule of thumb governing whether classical or non-classical behavior should prevail?
Solving the potential barrier smoothness conditions for relationships among the coefficients giving the reflection and transmission probabilities, usually involves rather messy algebra. However, there is a special case than can be done fairly easily, through requiring a slight departure from the standard solutions used in the chapter. Suppose the incident particles’ energyis precisely.
(a) Write down solutions to the Schrodinger Equation in the three regions. Be especially carefull in the region. It should have two arbitrary constants and it isn’t difficult – just different.
(b) Obtain the smoothness conditions, and from these find.
(c) Do the results make sense in the limit?
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