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Unformatted text preview: 2/21/11 1 PHYS 360 Quantum Mechanics Mon Feb 22, 2010 Lecture 17: What if the square well isn't innitely deep? (Bound states and scaering states) Recording? 1 HW 5 due on Monday Mar 1: Ch 2, #23, 30, 34, 35 d 2 k dx 2 = k 2 k , where k 2 mE k h For V=0, the meindependent Schrdingers equaon becomes: The eigenfuncons are: k x ( ) = Ae ikx + Be ikx The eigenstates are: k x , t ( ) = Ae i kx hk 2 2 m t However, these cannot be normalized: k * k dx + = A 2 dx = A 2 ( ) + 1 even though they are good eigenstates of denite energy. Hence, a free parcle cannot be in a denite energy state. The plan is to construct physical wave funcons out of unphysical (but mathemacally sound) eigenstates. x , t ( ) = 1 2 k ( ) e i kx hk 2 2 m t dk + ( x , t ) = c n n = 1 n ( x ) e iE n t / i .e. c n 1 2 ( k ) dk Note: this is based on the summaon we used for discrete states, but extended to an integral for connuous states. 4 How we solve for freeparcle wave funcons: 1. You are given a potenal (V(x)=0)....
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- Spring '11