sec9 - IX. The Wigner-Eckart theorem W e will now touch on...

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IX. The Wigner-Eckart theorem We will now touch on a result that relates very deeply to the rotational symmetry of space. However, its practical consequences are somewhat limited, and so we will only go over it in faint detail. The basic thing that we observe is that simply looking at how quantum states behave under rotation gave us a terrific amount of insight into the quantum theory of angular momentum. One can ask the corresponding question about operators: how do quantum operators behave when we rotate the system and what do we learn about operators in general (and most importantly, their matrix representations!) by studying their rotational character? We will find many connections to our results for the addition of angular momenta. a. Spherical Tensors Key to the statement of the Wigner-Eckart theorem is the definition of spherical tensor operators. This is rendered quite difficult by the fact that most chemists and physicists do not know what a spherical tensor is (never mind the operator part). This is just a geometrical concept, and once again we will find that the transition to quantum mechanics is trivial; all the “weirdness” is classical. Under rotation, a classical observable T changes as R T ( p r ) ⎯ → T ( R ( θ ) R r ( ) p ) , , n n R where “ ⎯ ⎯→ ” indicates the act of rotation and R ( ) is a matrix n rotates vectors by an angle θ about a unit vector n, as in the section on angular momentum. Now, the interesting fact is that there are certain groups of observables that are related to one another by rotation. Further, one finds that these groups always have an odd number of members! Any such group of observables with 2 k + 1 elements is said to compose a spherical tensor of rank k
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This note was uploaded on 11/28/2011 for the course CHEM 5.74 taught by Professor Robertfield during the Spring '04 term at MIT.

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sec9 - IX. The Wigner-Eckart theorem W e will now touch on...

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