PS7 - Physics 115A Problem Set 7 Due Tuesday November 10...

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Physics 115A, Problem Set 7 Due Tuesday, November 10 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths Section 2.6, 3.1. If you have not studied linear algebra recently, please read the Appendix in Griffiths carefully. 1 From the Square Well to the Delta Well We can think of the Dirac delta function as the limiting case of a rectangle of unit area as the height goes to infinity and the width goes to zero, so we should be able to obtain the physics of the delta function well as a limiting case of the finite square well. 1. Show that the delta function well is a “weak potential” in the sense that it is the z 0 0 limit of a finite square well. 2. Determine the bound state energy for the delta function potential by treating it as a limit of a finite square well. 3. Determine the transmission coefficient T for the delta function potential by treating it as a limit of a finite square well. 2 The Square Barrier Let’s work out the transmission coefficient for a rectangular barrier defined by V ( x ) = V 0 - a x a 0 | x | > a for V 0 > 0. There are three possible cases of interest: E < V 0 ; E = V 0 ; and E > V 0 , and the wavefunction inside the barrier is different in each case. 1. Find T - 1 for E < V 0 2. Find T - 1 for E = V 0 3. Find T - 1 for E > V 0 (Hint: This one is easily related to the finite square well.) 1
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3 Odd Wells
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