hwsol8 - Physics 31 Spring 2008 Solution to HW#8 In region...

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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 0 V = 0 x - L /2 0 + L /2 E I II (a) Write down the explicit form of the time-independent 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 time-independent 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 ¯ 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 solutions, so we take the sine term. That is, we set A = 1 and B = 0, and we won’t worry about normalization.
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