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Unformatted text preview: Physics 3220 Quantum Mechanics 1 Spring 2009 Problem Set #8 Due Wednesday, March 11 at 9am Problem 8.1 : Qualitative methods for stationary states. (20 points) a) The potential energy for a particle is given by V , x < , V ( x ) = 0 , < x < a, (1) V / 2 , a < x < 2 a, V , 2 a < x. Sketch this potential. Assume V and a have been chosen so that 0 < E 1 < V / 2 < E 2 < V for the energies E 1 , E 2 of the first two stationary states. Without actually solving the TISE, draw two separate sketches, one for the ground state u 1 ( x ) with energy E 1 , and one for the first excited state u 2 ( x ) with energy E 2 . Comment on the features of the curves everywhere, including things like wavelength, curvature and amplitude, and describe what happens at x = 0 ,a, 2 a , using your qualitative knowledge of wavefunctions. (A sketch with no explanation will receive little credit!) b) For the asymmetric well pictured below, sketch the states u 1 ( x ) (ground state), u 2 ( x ) (first excited state) and some larger u n ( x ) (like n = 10 or so). Explain in words the relevant important features of the wavefunctions, including things like wavelength, curvature and amplitude. Also, describe a physical example of a potential energy function that looks like this.describe a physical example of a potential energy function that looks like this....
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