Week #3 (Due Sunday, Feb.1)

 

1. Read the following :

 

Cramer Chapter 2, Sections 2.1,2.2,2.3 and 2.6

 

A discussion question will be posted on the Oncourse site Wednesday.

 

2. Consider the potential energy expression used to model the umbrella inversion of ammonia

 

 

1. Find the extrema for this potential

                                                     i.     Extrema will occur when the first derivative is zero or

 

This is zero when x=0 or 1,-1

 

2. Determine, analytically, if the extrema are minima or maxima

                                                     i.     An extremum is a minimum if the second derivative is positive or a maximum if the second derivative is negative

 

                                    Substituting the values of x at the extrema we find:

                                               

 

This surface represents the umbrella motion of ammonia. In particular, x=0, corresponds to the top of the barrier to the inversion between the two trigonal pyramids, which is not allowed, classically but does occur due to quantal tunneling

                                               

                                               

 

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