HW7 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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Physics 3220 – Quantum Mechanics 1 – Fall 2008 Problem Set #7 Due Wednesday, Oct 15 at 2pm Problem 7. 1 Qualitative methods. [20 pts] A) The potential energy V(x) for a particle is given by: V ( x ) = V 0 for x < 0 0 for 0 < x < a V 0 /2 for a < x < 2a V 0 for 2a < x Sketch this potential. Assume “V 0 ” and “a” have been chosen so that 0 < E 1 < V 0 /2 < E 2 < V 0 . Now draw two separate sketches, one for the ground state (call it u 1 (x), with energy E 1 ), another one for the 1 st excited state (u 2 (x), with energy E 2 ). Think carefully about, and comment on, the features of the curves everywhere, including what happens at x=0, a, and 2a. (As always, your job is to convince us that you understand. A sloppy sketch without explanation may receive little or no credit.) B) Let’s make the potential simpler, just a “finite square well”: V ( x ) = + V 0 for x < 0 0 for 0 < x < a +V 0 for x > a Derive an approximate simple expression for the number of bound states in the well, in terms of a and V 0 . Your expression does NOT have to be exact, in fact no long or messy calculations are permitted, just make a quick “intelligent crude guess” ( Hint: Just think about the energies in an infinite square well! ) Please also guess/briefly discuss whether your estimate is more likely to over estimate, or under estimate, the correct answer. C) Sketch a few energy eigenstates u n (x) for the asymmetric potential well shown below. In particular, sketch n = 1 (ground state), n = 2, and some large n (like n = 10 or so). ALSO , please invent/ describe a physical example of a system that has a potential energy graph like this. In your sketches, clearly explain in words the important features of the wavefunctions. (Don’t forget to discuss “wavelength”, “amplitude”, and “curvature” behaviors) (Note: You can check yourself with phet.colorado.edu, go to the “Quantum bound states” sim. Another more complicated sim is available at www.falstad.com/qm1d/ which might be useful for part a, though it’s considerably trickier to use than Phet ) Important: we are not interested in having you copy sim outputs – we want your basic physics explanations. (Just use sims to check you haven’t missed any key features!) Problem
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HW7 - Physics 3220 Quantum Mechanics 1 Fall 2008 Problem...

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