Chapter11_marked - Chapter 11 Vibrational motion and...

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10/3/13 1 Chapter 11 Vibrational motion and infrared spectroscopy Simple Harmonic Oscillator: Classical theory Classically, a mass attached to a spring will vibrate between A and -A. F = " k x F = m a = m dv dt = m d 2 x dt 2 m d 2 x dt 2 = " k x x = A sin " t d 2 x dt 2 = d dt d dt A sin " t # $ % & ' ( = d dt ) A " cos " t ( ) = ) A " 2 sin " t = ) " 2 x m " # 2 x ( ) = " k x or # = k m Simple Harmonic Oscillator: Quantum theory E " ( x ) = # ! 2 2 m d dx 2 2 " ( x ) + V ( x ) " ( x ) F = " k x V = " Fdx = " " kx ( ) # # dx = k x dx = k 2 # x 2
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10/3/13 2 4. E ! ( x ) = ! ! 2 2 m d 2 dx 2 ! ( x ) + k 2 x 2 " # $ % & ' 2 ! ( x ) 3. E ! ( x ) = ! ! 2 2 m d 2 dx 2 ! ( x ) 2. E ! ( x ) = ! ! 2 2 m d 2 dx 2 ! ( x ) + k x ! ( x ) 1. E ! ( x ) = ! ! 2 2 m d 2 dx 2 ! ( x ) + k 2 x 2 ! ( x ) Schrodinger’s equation for the harmonic oscillator is V = k 2 x 2 E " ( x ) = # ! 2 2 m d 2 dx 2 " ( x ) + k 2 x 2 " ( x ) " ! 2 2 m d 2 dx 2 # ( x ) = E " k 2 x 2 $ % & ' ( ) # ( x ) Consider the limit x " # then E << k 2 x 2 E $ k 2 x 2 % $ k 2 x 2 " ! 2 2 m d 2 dx 2 # ( x ) = " k 2 x 2 # ( x ) Schrodinger’s equation for the simple harmonic oscillator
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10/3/13 3 ! ( x ) = e ! bx 2 ! ! 2 2 m d 2 dx 2 ! ( x ) = ! k 2 x 2 " # $ % & ' ! ( x ) E " ( x ) = # ! 2 2 m d 2 dx 2 " ( x ) + k 2 x 2 " ( x ) " v ( x ) = N v H v e # x 2 / 2 $ 2 " = ! 2 mk # $ % & ' ( 1/ 4 , v = 0,1, 2,... N v = 1 " # 1/ 2 2 v v ! v! = v (v-1) (v-2) (v-3)…(1) v H v (x) 0 1 1 2(x/ ) 2 4(x/ ) 2 - 2 3 8(x/ ) 3 – 12(x/ ) Hermite equation
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