486_HW5 - Physics 486 FA07 Homework Set 5 Due Oct 4 2007 1 Based on Griffiths 2.22 A free particle has the initial Gaussian wave function Ψ(x,0 =

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Unformatted text preview: Physics 486 FA07 Homework Set 5 Due: Oct 4, 2007 1) Based on Griffiths 2.22 . A free particle has the initial Gaussian wave function Ψ (x,0) = A e-ax ² , where A and a are constants (a is real and positive). (a) Determine the constant A. (b) Find the time-dependent wavefunction Ψ (x,t). Hint: Integrals of the function e- (ax ² +bx) (limits - ∞ to + ∞ ) can be performed by “completing the square”: Let y = √ a (x+(b/2a)) and note that ax ² +bx = y ²- (b ² /4a). (c) Find | Ψ (x,t)| ² . Express your answer in terms of the quantity w = [a/(1+(2at ℏ /m) ² )] 1/2 . Sketch | Ψ (x,t)| ² as a function of x at t = 0, and once again at some very large time t. Qualitatively, what happens to | Ψ | ² as time goes on? 2) Double well potential. Consider the “double square well” potential model of an H 2 or NH 3 molecule. Suppose the depth, V , and the width, a , are fixed and large enough that several bound states occur. The separation, b , can be varied....
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This note was uploaded on 09/20/2010 for the course PHYS 486 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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486_HW5 - Physics 486 FA07 Homework Set 5 Due Oct 4 2007 1 Based on Griffiths 2.22 A free particle has the initial Gaussian wave function Ψ(x,0 =

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