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Chapter 6: Q16E (page 224)

A particle moving in a region of zero force encounters a precipice---a sudden drop in the potential energy to an arbitrarily large negative value. What is the probability that it will “go over the edge”?

Short Answer

Expert verified

I will definitely reflect for U0tending to infinite.

Step by step solution

01

Given known parameters

T=4E(E-U0)(E+E-U0)2

02

Solution

As U0tends to ∞, the transmission coefficient tends to 0

Thus reflection is definitely possible.

03

Explanation and Conclusion

We might suppose that there should be a smooth match to an answer within the region beyond the downward step. But the actual fact that the P.E. is infinite means the derivative is discontinuous at the drop. The wave function is solely an undulation to the left of the step, zero at the step, and nil beyond the step.

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Most popular questions from this chapter

Question: Obtain equation (6.18) from(6.16) and (6.17).

Solving the potential barrier smoothness conditions for relationships among the coefficients A,B and Fgiving 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’ energyEis preciselyU0.

(a) Write down solutions to the Schrodinger Equation in the three regions. Be especially carefull in the region0<x<L. It should have two arbitrary constants and it isn’t difficult – just different.

(b) Obtain the smoothness conditions, and from these findR and T.

(c) Do the results make sense in the limitL→∞?

The diagram below plots ω(k) versus wave number for a particular phenomenon. How do the phase and group velocities compare, and do the answer depend on the central value of k under consideration? Explain.

A method for finding tunneling probability for a barrier that is "wide" but whose height varies in an arbitrary way is the so-called WKB approximation.

T=exp[−2ℏ∫122m(U(x)−E)dx]

Here U(x) is the height of the arbitrary potential energy barrier.Whicha particle first penetrates at x=0 and finally exits at x=L. Although not entirely rigorous, show that this can be obtained by treating the barrier as a series of rectangular slices, each of width dx (though each is still a "wide" barrier), and by assuming that the probability of tunneling through the total is the product of the probabilities for each slice.

Suppose the tunneling probability is10-12for a wide barrier when E is 1100U0

(a) About how much smaller would it be if ’E’ were instead 11000U0?

(b) If this case does not support the general rule that transmission probability is a sensitive function of E, what makes it exceptional?
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