385lec13

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

This preview shows pages 1–2. Sign up to view the full content.

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 ± .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## 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.

### Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online