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HW11 - University of Colorado Department of Physics...

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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
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