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Unformatted text preview: Physics 31 Spring, 2008 Solution to HW #8 Problem A For this problem, you are to work out the so lution for the wave function in a finite square well, for the case of an odd wave function. (In class, we did the case of an even wave function.) V = V V = 0 xL /2 + L /2 E I I I (a) Write down the explicit form of the timeindependent Schroedinger equation in region I. Let ψ I ( x ) = sin kx , where k 2 = 2 mE/ ¯ h 2 , and explain why this ψ I ( x ) is the desired solution to the Schroedinger equation. (b) Write down the explicit form of the timeindependent Schroedinger equation in region II, and verify that ψ II ( x ) = C exp( αx ) + D exp( − αx ) is a solution to this equation. What is α ? (c) Write down the equations for the continuity of the wave function and its derivative at the point x = L/ 2. Solve the equations to find the condition that must be satisfied in order to have C = 0. Explain why there is a physical solution only for certain values of the energy E . In region I, the Schroedinger equation is µ − ¯ h 2 2 m d 2 dx 2 + 0 − E ¶ ψ = 0 , where the 0 is the value of V ( x ) in region I. The general solution to this equation is ψ I ( x ) = A sin kx + B cos kx, where k 2 = 2 mE/ ¯ h 2 . In class we considered the even so lutions and took the cosine term. Now we want the odd...
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This note was uploaded on 08/06/2008 for the course PHYS 31 taught by Professor Hickman during the Spring '08 term at Lehigh University .
 Spring '08
 Hickman
 mechanics, Work

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