(a)
Prove that the reflection coefficient for the given condition is
(a)
Answer to Problem 59CP
Proof for the reflection coefficient for the given condition is
Explanation of Solution
Write the Schrodinger’s equation.
Here,
The solutions of the equation I for the region I in the above figure is
The solutions of the equation I for the region II in the above figure is
Here,
Check the solution for the region I satisfies the equation I.
The above equation will be true only if the
Here,
Rewrite the above equation to find the value of
Therefore the equation I is satisfied for the region I.
Check the equation I for the region II.
The above equation will be true only if the
Rewrite the above equation to find the value of
Therefore the equation I is satisfied for the region II.
Then apply the boundary conditions such as matching the function and derivatives at x=0.
From the above equations,
Write the equation for probability.
Conclusion:
Substitute equation VI in VII.
Therefore, the proof for the reflection coefficient for the given condition is
(b)
Find the probability of particle being reflected.
(b)
Answer to Problem 59CP
The probability of particle being reflected is
Explanation of Solution
Write the equation for ration of
Conclusion:
Substitute
Substitute
Therefore, the probability of particle being reflected is
(c)
Find the probability of particle being transmitted.
(c)
Answer to Problem 59CP
The probability of particle being transmitted is
Explanation of Solution
Write the equation for probability of transmitted.
Conclusion:
Substitute
Therefore, the probability of particle being transmitted is
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Chapter 41 Solutions
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