Lecture16 - FriFeb18,2011 Lecture16...

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2/21/11 1 PHYS 360 Quantum Mechanics Fri Feb 18, 2011 Lecture 16: Time‐dependence of wave packets…. HW 4 due on Monday: Ch 2, #13, 14, 19, 22 Recap results from last Rme for a free parRcle: d 2 ψ k dx 2 = k 2 ψ k , where k 2 mE k h For V=0, the Rme‐independent Schrödinger’s equaRon becomes: The eigenfuncRons are: ψ k x ( ) = Ae ikx + Be ikx The eigenstates are: Ψ k x , t ( ) = Ae i kx hk 2 2 m t However, these cannot be normalized: Ψ k * Ψ k dx −∞ + = A 2 dx = A 2 ( ) −∞ + 1 … even though they are good eigenstates of definite energy. Hence, a free parRcle cannot be in a definite energy state. The plan is to construct physical wave funcRons out of unphysical (but mathemaRcally sound) eigenstates. Ψ x , t ( ) = 1 2 π φ k ( ) e i kx hk 2 2 m t dk −∞ + Ψ ( x , t ) = c n n = 1 ψ n ( x ) e iE n t / i .e. c n 1 2 π ϕ ( k ) dk Note: this is based on the summaRon we used for discrete states, but extended to an integral for conRnuous states. 5 How we solve for free‐parRcle wave funcRons: 1. You are given a potenRal (V(x)=0).
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