# 21 - Physics 6210/Spring 2007/Lecture 21 Lecture 21...

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Physics 6210/Spring 2007/Lecture 21 Lecture 21 Relevant sections in text: § 3.1, 3.2 Rotations in three dimensions We now begin our discussion of angular momentum using its geometric interpretation as the generator of rotations in space. I should emphasize at the outset that our discussion can be a little confusing because we will be studying vectors and linear transformations in 2 distinct spaces: (i) the 3-d (Euclidean) space we live in, and (ii) the Hilbert space of quantum state vectors. The 3-d rotations are, of course, going to be related to correspond- ing transformations on the space of quantum states, but it is not too hard to get mixed up about which space various quantities are associated with. So watch out! We begin by summarizing some elementary results concerning rotations in three di- mensions. This part of the discussion is completely independent of quantum mechanical considerations. Until you are otherwise notiﬁed, everything we do will only refer to prop- erties of rotations of observables in the 3-d space live in. A vector observable for some physical system, ~ V , responds to a rotation according to a (special) orthogonal transformation: ~ V R ~ V . Here R is a linear transformation of 3-d vectors such that ( R ~ V ) · ( R ~ W ) = ~ V · ~ W. Evidently, magnitudes of vectors as well as their relative angles are invariant under this transformation. If you represent vectors ~ V , ~ W as column vectors V , W relative to some Cartesian basis, the dot product is ~ V · ~ W = V T W = W T V.

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## This note was uploaded on 02/18/2012 for the course PHYSICS 6210 taught by Professor M during the Spring '07 term at AIU Online.

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21 - Physics 6210/Spring 2007/Lecture 21 Lecture 21...

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