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Unformatted text preview: Physics 731 Assignment #4, Solutions 1. The odd parity eigenfunctions of the finite square well potential, V = , | x | < a V , | x | > a, (1) for regions I ( x <- a ), region II (- a < x < a ), and region III ( x > a ) take the form I =- Fe x , II = B sin kx, III ( x ) = Fe- x , (2) in which k = 2 mE/ h and = p 2 m ( V- E ) / h . Continuity of and d/dx at x = a leads to the conditions F = B sin ka e a (3)- a = ka cot ka. (4) Defining as usual = ka and = a , we have the conditions =- cot , 2 + 2 = R 2 , (5) in which R = 2 mV a/ h . Using either graphical or numerical methods, it is straightforward to see that for there are no solutions for R < / 2 ( i.e. , V < 2 h 2 / (8 ma 2 ) , there is one solution for / 2 < R < 3 / 2 , and there are two solutions for 3 / 2 < R < 5 / 2 , and so on. Therefore, as V , there are no odd parity bound states, and for V , there are an infinite number of bound states (as expected), with k = n/ (2 a ) for even values of n . Together with Eq. ( 4 ), the normalization condition for the eigenfunctions, Z a- a | B | 2 sin 2 kxdx + 2 Z a | F | 2 e- 2 x dx = 1 , (6) yields the following result for B : | B | = 1 p a (1 + 1 / ( a ) , (7) which reduces to the infinite square well case for V . The complete eigenfunctions for the odd parity states thus are given by I =- 1 p a + 1 / e ( x + a ) , II = 1 p a + 1 / sin kx, III ( x ) = 1 p a + 1 / e- ( x- a ) . (8) 2. We are asked to analyze the bound states of the above finite square well potential with 2 mV a 2 h 2 ! 1 2 = 2 . (9) The bound states, which have E < V , can be classified by even/odd parity. For the even parity states, the conditions which determines the bound state energies are tan = , 2 + 2 = R 2 = 2 mV a 2 h 2 , (10) 1 where = ka = 2 mEa/ h and = a = p 2 m ( V- E ) a/ h . Here we have R = 2 . To determine the bound state energies, one way to proceed is as we did in class, e.g. plot the intersection of tan and the circle 2 + 2 = R 2 . Another option is to form the ratio f ( ) = p 4- 2 tan , (11) in which case the allowed bound states are given by...
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- Fall '09