385lec13 - PHYS 385 Lecture 13 - Harmonic oscillator in 1D...

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PHYS 385 Lecture 13 - Harmonic oscillator in 1D 13 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 13 - Harmonic oscillator in 1D What's important : Harmonic oscillator in 1D Hermite polynomials Text : Gasiorowicz, Chap. 5 Harmonic oscillator in 1D As a generic system, the harmonic oscillator V ( x ) = kx 2 /2 (1) has widespread application, particularly as an approximation for more functionally complex systems near their ground state energy. For example, in Lec. 2 of PHYS 211, the effective spring constant k is extracted from an approximation to the Morse potential in atomic physics. With Eq. (1), the time-independent Schrödinger equation in one dimension reads - h 2 2 m d 2 dx 2 u ( x ) + 1 2 kx 2 u ( x ) - Eu ( x ) = 0 To simplify the notation, recall the angular frequency of simple harmonic motion = ( k / m ) 1/2 -> k = m 2 , (2) which permits (1) to be written as d 2 dx 2 u ( x ) + 2 mE h 2 u ( x ) - m 2 2 h 2 x 2 u ( x ) = 0. Defining = 2 mE / h 2 2 = m 2 2 / h 2 (3) we have d 2 dx 2 u ( x ) + u ( x ) - 2 x 2 u ( x ) = (4) To find the solutions for u ( x ), let's first look at the asymptotic regions where x ± . Then, we can neglect in Eq. (4) and just solve d 2 u / dx 2 = 2 x 2 u for x ± . (5) Eq. (5) has the approximate solution u ( x ) exp(± x 2 /2) for x ± .
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This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.

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385lec13 - PHYS 385 Lecture 13 - Harmonic oscillator in 1D...

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