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pset03

# pset03 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum...

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2 1 V x V x V x MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum Mechanics I Fall , 2002 Professor Robert W. Field Problem Set #3 DUE : At the start of Lecture on Friday , September 27. Reading : Merzbacher , pages 113-134 Note : We are going to return to some of these problems once we learn how to use perturbation theory. Problems : 1. Airy Functions , piecewise linear potentials For the following 5 potentials , find the energies of the 5 lowest energy eigenstates. 1 3 h 2 2 / C Let 2 m = × 0 6 erg. for all potentials. You are free to choose specific values, as needed to make computer programs work, for all unspecified constants (m,L). A. Vee-box. V(x) = C|x| C > 0 ( optional )B. Vee-box plus δ -function (a symmetric double minimum potential) V(x) = C|x| + a δ (x) a = 10C , C > 0 C. Vee-bottom box . ( ) = ∞ | x | > L / 2 CL ( ) = Cx 0 < x L / 2 4 CL ( ) = − Cx L / 2 x < 0 4 Warning: This is tricky. You will need to use both A i and B i .

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V x V x V x V x V x Chemistry 5.73 Page 2 Problem Set #3 D. Inverted Vee-bottom infinite box. ( ) = ∞ | x | > L / 2 CL ( ) = − Cx + 0 < x L / 2 4 CL ( ) = Cx + L / 2 x < 0 4 E. Slant-bottom infinite box. ( ) = ∞ | x | > L / 2 ( ) = Cx | x | L / 2 ( optional )F. Compare the energies from parts A and B and discuss in terms of tunneling . G. Compare the energies from parts C , D , and E to the energies of an ordinary infinite box , V( x ) = for | x | > L /2 , V( x ) = 0 for | x | L /2. All three boxes have the same average depth. If the energies are not identical , explain. If they are identical , perhaps the WKB quantization condition (even though it is inapplicable here for several reasons) , applied at E = CL/2, will provide an explanation.
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pset03 - MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum...

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