4604_Exam3_fa06 - PHY4604 Fall 2006 Final Exam PHY 4604...

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PHY4604 Fall 2006 Final Exam Department of Physics Page 1 of 6 PHY 4604 Final Exam Take Home Exam (Total Points = 100) You may use books, notes, internet, and computers. However, you are not allowed to get help from other people. You must do the work yourself! You can spend as much time on it as you want, but you must return the exam on Wednesday December 6 ( in class ). If I do not receive your exam by 10:30am on Wednesday December 6, you will receive a zero on the exam. Problem 1 (25 points): Consider the following one-dimensional potential: + = 0 0 ) ( V x V L x L x L x < < 0 where V 0 is a positive (real) constant. (A) Suppose that particles with mass m and energy E > 0 enter from the left in region 1 and travel to the right and encounter the potential V(x) . Let 2 2 2 2 ε mL E h = and E V r 0 1 + = . (1) (4 points) Calculate the probability that particles entering from the left in region 1 will be reflected back ( i.e. calculate the ratio of the probability flux in region 1 traveling to the left to the incident flux in region 1 traveling to the right, 1 1 / j j P R r s = ). Express your answer in terms of ε and r. (2) (4 points) Calculate the probability that particles traveling to the right in region 2 will be reflected back ( i.e. calculate the ratio of the probability flux in region 2 traveling to the left to flux in region 2 traveling to the right, 2 2 / j j R r s = ). Express your answer in terms of ε and r. (3) (6 points) Calculate the ratio of the probability flux in region 2 traveling to the right to the incident flux in region 1 traveling to the right, 1 2 / j j P r r = ). Express your answer in terms of ε and r. What are the maximum and minimum values of P( ε ) (express your answer in terms of r)? Evaluate the maximum and minimum values of P( ε ) for the case V 0 = 3E . Explain what is going on. (B) Now consider the case E < 0 and look for possible bound states. Let 2 2 2 2 mL E h = and 2 2 2 0 2 α mL V h = . (1) (5 points) How many bound states are there if α = π /4 ? What is the ground state energy ( i.e. what is ε for the ground state)? (2) (6 points) How many bound states are there if α = 11 π /4 ? What is the ground state energy ( i.e. what is ε for the ground state)?
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This note was uploaded on 05/29/2011 for the course PHY 4064 taught by Professor Fields during the Spring '07 term at University of Florida.

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4604_Exam3_fa06 - PHY4604 Fall 2006 Final Exam PHY 4604...

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