Lecture18 - CHEM 356, Lecture 18, Fall 2009 1 Quantum...

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Unformatted text preview: CHEM 356, Lecture 18, Fall 2009 1 Quantum Mechanics of the Rigid Rotor : the hamiltonian for the motion of a rigid rotor can be obtained as rot = 2 2 . The orientation of the rigid rotor is completely specified by the two angles and . If we designate the rigidrotor state functions by ( , ), then the Schr odinger equation for this rigidrotor motion is rot ( , ) = rot ( , ) , or 1 2 J 2 ( , ) = rot ( , ) . Because we are examining an orbital type of motion, we can employ the differential operator form to represent J 2 , thereby obtaining the differential equation 2 2 [ 1 sin ( sin ) + 1 sin 2 2 2 ] ( , ) = rot ( , ) . By multiplying both sides of this equation by sin 2 , and defining a parameter via 2 rot 2 , we obtain the partial differential equation (PDE) governing ( , ), namely sin ( sin ) + 2 2 + sin 2 = 0 . The equation for ( , ) can be solved by the method of separation of variables. To do so, we write ( , ) = ( )( ), and substitute this form for into the PDE, to obtain sin ( ) ( sin ) + sin 2 + 1 ( ) 2 2 = 0 . Note that this equation contains three terms, two depending solely upon , the third solely upon . CHEM 356, Lecture 18, Fall 2009 2 Aside: If we recall that ? = , we see that the third term in our equation is just 1 ( ) 2 ? 2 , and that the functions ( ) = e , = 0 , 1 , 2 , , with determined from normalization to be 1/ 2 , are the eigenfunctions of ?...
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This note was uploaded on 02/28/2011 for the course CHEM 356 taught by Professor Prof.iaskjd during the Fall '09 term at Waterloo.

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Lecture18 - CHEM 356, Lecture 18, Fall 2009 1 Quantum...

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