Lecture16.pptx

Lecture16.pptx - 2/21/11 1 PHYS 360 Quantum Mechanics Fri...

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Unformatted text preview: 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 ¡me for a free par¡cle: 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....
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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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Lecture16.pptx - 2/21/11 1 PHYS 360 Quantum Mechanics Fri...

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