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homework3 - 1 Problem set 3 Square well potential Due Sept...

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1 Problem set 3 - Square well potential Due: Sept. 30, 2011, in class Posted on Sept. 23, 2011 1. Bohr Sommerfeld quantization of a particle in a box Assume we did not know about Schr¨ odinger formalism and wanted to solve the particle in a box problem. According to Bohr-Sommerfeld quantization we should look for periodic motion and quantize the action. Assume the particle is a classical object (like a tennis ball) which starts its motion at position x 0 [0 , L ] with some momentum p . (a) Will the particle go through periodic motion? (b) If so - what is the time it takes to complete a cycle? What distance is traveled? (c) Calculate the classical action S = R pdx over one cycle (you can either integrate over position or change the integration variable to time like we did in lecture 8). (d) Assume the action is quantized in units of Plancks constant h . What are the allowed wave numbers k n ? What are the Energies E n ? (e) Is the Bohr-Sommerfeld a good approximation in this case?
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