{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# HW11 - University of Colorado Department of Physics...

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

University of Colorado, Department of Physics PHYS3220, Fall 09, HW#11 due Wed, Nov 4, 2PM at start of class 1. Analytic solution of the harmonic oscillator (Total: 20 pts) In this problem we go through the analytic solution of the time-independent Schr¨ odinger equation for the harmonic oscillator - ~ 2 2 m d 2 χ ( x ) dx 2 + 1 2 2 x 2 χ ( x ) = ( x ) (1) a) It is convenient to simplify the problem by introducing the two dimensionless variables ξ = p ~ x and K = 2 E ~ ω Show that the time-independent Schr¨ odinger equation can be written as d 2 χ ( ξ ) 2 = ( ξ 2 - K ) χ ( ξ ) b) Show that in the limit | x | → ∞ the above equation can be approximated as d 2 χ ( ξ ) 2 ξ 2 χ ( ξ ) and an approximate solution is given by χ ( ξ ) A exp( - ξ 2 / 2) + B exp(+ ξ 2 / 2) What is the constraint needed to be sure that χ ( ξ ) can be normalized? (Don’t actually try to normalize the function.) c) The original equation turns out to simplify if we extract the asymptotic (large | x | ) behavior of χ and solve for what’s left. Accordingly, we define H

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.

{[ snackBarMessage ]}