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Lecture10 - Phys 436 Modern Physics Lecture 10 Jan 31 2013...

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Phys 436 – Modern Physics Lecture 10 Jan. 31, 2013 Instructor: David Cooke Molecular vibraDons Molecular rotaDons 1
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Assignment #2 posted Due date Feb. 7 th , (in class at the beginning of class.)
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Molecular vibraDons dissociaDon energy V(R) R R 0 Last Dme we saw how the potenDal energy of the ions displays a minimum at R=R 0 R-­૒R 0 V(R) Quantum mechanical harmonic potenDal You’ll remember the soluDon to the harmonic oscillator from Quantum Mechanics
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Harmonic potenDal R-­૒R 0 Taylor expand potenDal about the minimum: V ( R ) = V ( R 0 ) + dV ( R ) dR ( R ! R 0 ) + 1 2 d 2 V ( R 0 ) dR 2 ( R ! R 0 ) 2 + ... = 0 V ( R ) ! 1 2 d 2 V ( R 0 ) dR 2 ( R " R 0 ) 2 ! 0 V(R) ! ! 2 2 m d 2 dx 2 + 1 2 kx 2 " # $ % & ' ( ( x ) = E ( ( x )
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Recognize this problem from QM m m m m F = ! dV dx = ! kx x ( t ) " cos ( # t ) x ! ! 2 2 m d 2 dx 2 + 1 2 kx 2 " # $ % & ' ( ( x ) = E ( ( x ) Quantum mechanical Classical: Hooke’s law V = 1 2 kx 2 SoluDon done in Phys 446 QM ! = k m ! n ( x ) = m " # ! $ % & ' ( ) 1/4 1 2 n n ! H n ( * ) e + * 2 /2 E n = n + 1 2 $ % & ' ( ) ! " Energies are quanDzed and equally spaced ! = m " ! x ! E = ! "
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SoluDons ! n ( x ) = m " # ! $ % & ' ( ) 1/4 1 2 n n !
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