LectureCh14.pdf

# N we obtain mn d r e dr a m a n h m h n e 2 d using

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n ( ? ) , we obtain ? mn = ( d ? ( r e ) dr ) A m A n -∞ H m ? H n e - ? 2 d ? Using recursive relation, ? H n = 1 2 H n + 1 + nH n - 1 , we obtain ? mn = ( d ? ( r e ) dr ) A m A n [ 1 2 -∞ H m H n + 1 e - ? 2 d ? + n -∞ H m H n - 1 e - ? 2 d ? ] To simplify expression we rearrange A m A n -∞ H m ( ? ) H n ( ? ) e - ? 2 d ? = 𝛿 m , n to -∞ H m ( ? ) H n ( ? ) e - ? 2 d ? = 𝛿 m , n A m A n Substitute into expression for ? nm gives ? mn = ( d ? ( r e ) dr ) [ 1 2 A n A n + 1 𝛿 m , n + 1 + n A n A n - 1 𝛿 m , n - 1 ] P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 24 / 26

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Harmonic Oscillator Recalling A n 1 2 n n ! 𝜋 1 2 we finally obtain transition dipole moment for harmonic oscillator ? mn = ( d ? ( r e ) dr ) [ n + 1 2 𝛿 m , n + 1 + n 2 𝛿 m , n - 1 ] For absorption, m = n + 1 , transition is n n + 1 and ? mn 2 gives R n n + 1 = 𝜌 ( ? mn ) ? mn 2 6 𝜖 0 2 = 𝜌 ( ? mn ) 6 𝜖 0 2 ( d ? ( r e ) dr ) 2 n + 1 2 For emission, m = n - 1 , transition is n n - 1 and ? mn 2 gives R n n - 1 = 𝜌 ( ? mn ) ? mn 2 6 𝜖 0 2 = 𝜌 ( ? mn ) 6 𝜖 0 2 ( d ? ( r e ) dr ) 2 n 2 P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 25 / 26
Harmonic Oscillator Selection rule for harmonic oscillator is Δ n = ± 1 . Also, for allowed transitions ( d ? ( r e )∕ dr ) must be non-zero. For allowed transition it is not important whether a molecule has permanent dipole moment but rather that dipole moment of molecule varies as molecule vibrates. In later lectures we will examine transition selection rules for other types of quantized motion, such as quantized rigid rotor and orbital motion of electrons in atoms and molecules. P. J. Grandinetti (Chem. 4300) Radiating Dipoles in Quantum Mechanics Oct 27, 2017 26 / 26
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• Fall '13

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