Lecture17.pptx

Lecture17.pptx - 1 PHYS 360 Quantum Mechanics Mon Lecture...

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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 in¡nitely deep? (Bound states and sca¢ering states) Recording? 1 HW 5 due on Monday Mar 1: Ch 2, #23, 30, 34, 35 d 2 ψ k dx 2 = − k 2 ψ k , where k ≡ 2 mE k h For V=0, the £me‐independent Schrödinger’s equa£on becomes: The eigenfunc£ons 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 de¡nite energy. Hence, a free par£cle cannot be in a de¡nite energy state. The plan is to construct physical wave func£ons out of unphysical (but mathema£cally 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 summa£on we used for discrete states, but extended to an integral for con£nuous states. 4 How we solve for free‐par£cle wave func£ons: 1. You are given a poten£al (V(x)=0)....
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This note was uploaded on 04/23/2011 for the course PHYS 360 taught by Professor Durbin,stephen during the Spring '11 term at Purdue.

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Lecture17.pptx - 1 PHYS 360 Quantum Mechanics Mon Lecture...

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