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3890ps4_09 - Physical Chemistry I Chem 3890 Fall 2009...

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Chem 3890 Physical Chemistry I Fall 2009 Chemistry 3890 Problem Set 4 Fall 2009 Due: in class, Friday, October 9 Last revised: October 4, 2009 Additions/corrections: Wavefunctions live in Hilbert space. D.J. Griffiths, Quantum mechanics , p94 Quantum phenomena do not occur in a Hilbert space; they occur in a laboratory. Asher Peres Reminder : For the Mathematica portions of the HW, you must submit a printout of an Evaluated notebook. It is a good idea to first Save your evaluated notebook in pdf format, then print the pdf file. Printouts of unevaluated notebooks will not be graded. Problem 1 In this problem we solve the time-independent Schr¨odinger equation for a particle in a box that has a step at the midpoint of the box. The Schr¨odinger equation is - planckover2pi1 2 2 m d 2 ψ ( x ) d x 2 + V ( x ) ψ ( x ) = ( x ) , (1) where m is the particle mass, E is a real parameter (the energy), and V ( x ) is the potential V ( x ) = x< 0 0 0 x a 2 ¯ V a 2 x a x>a (2) 1. Make a sketch of this potential for ¯ V > 0. 2. Show that the general solution for ψ ( x ) on the interval 0 x a/ 2 can be written as ψ ( x ) = A 1 sin( k 1 x ) + B 1 cos( k 1 x ) , (3) with k 1 2 mE/ planckover2pi1 . What is the boundary condition that ψ ( x ) satisfies at x = 0? Impose this BC on ψ ( x ). 3. There is a corresponding expression for ψ ( x ) on the interval a/ 2 x a , ψ ( x ) = A 2 sin( k 2 x ) + B 2 cos( k 2 x ) , (4) with k 2 radicalbig 2 m ( E - ¯ V ) / planckover2pi1 . (We take 0 ¯ V <E .) What is the BC that ψ ( x ) must satisfy at x = a ?
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