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3502_12_octo07_LG.1 - Chem 3502/5502 Physical Chemistry...

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Chem 3502/5502 Physical Chemistry II (Quantum Mechanics) 3 Credits Fall Semester 2009 Laura Gagliardi Lecture 12, October 7, 2009 Solved Homework We are allowed to know any one component of the total angular momentum to perfect accuracy. Although historically one nearly always speaks of the z component as being the one that is knowable (i.e., for which we can find eigenfunctions), this is a completely arbitrary labeling scheme. We might as easily name it x or y and the mathematics are identical. The Heisenberg uncertainty principle simply says that we cannot know more than one simultaneously. So, the eigenvalues of L x 2 are the same as those of L y 2 are the same as those of L z 2 —but we can only measure any one of these at any given time to arbitrary accuracy. In any case, we have generically for any coordinate q L q 2 ! = L q L q ! [ ] = L q m l h ! [ ] = m l h L q ! [ ] = m l h m l h ! [ ] = m l 2 h 2 ! where the quantum numbers m l can take on integer values from l to l where l is the quantum number for the square of the total angular momentum operator. Now, given what we just discussed above, one might think that it is impossible to obtain eigenvalues for L x 2 + L y 2 , but that’s not true. It’s not possible to have eigenvalues for each operator in the sum simultaneously, but it’s entirely OK to have an eigenvalue for the sum without knowing the two parts. To see this, note that L 2 = L x 2 + L y 2 + L z 2 so L 2 ! L z 2 = L x 2 + L y 2 and we already know that we can know both L 2 and L z 2 simultaneously! Thus, we have
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12-2 L x 2 + L y 2 ( ) ! = L 2 " L z 2 ( ) ! = l l + 1 ( ) h 2 " m l 2 h 2 [ ] ! = h 2 l l + 1 ( ) " m l 2 [ ] ! One can think of these eigenvalues as being the part that needs to be added to L z 2 in order to reach L 2 , but this additional component of the angular momentum can point in any direction within the plane defined by the z component having a constant value; that is, we don’t know the individual x and y components, so at best we know a circle of points around the z axis at which our angular momentum vector terminates. The angular momentum is said to “precess” about the z axis on this circle.
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