# 26 - Physics 6210/Spring 2007/Lecture 26 Lecture 26...

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Physics 6210/Spring 2007/Lecture 26 Lecture 26 Relevant sections in text: § 3.6, 3.7 Two spin 1/2 systems: observables We have constructed the 4-d Hilbert space of states for a system consisting of two spin 1/2 particles. We built the space from the basis of “product states” corresponding to knowing the spin along z for each particle with certainty. General states were, however, not necessarily products but rather superpositions of such. How are the observables to be represented as Hermitian operators on this space? To begin, let us consider the spin observables for each of the particles. Call them ~ S i = ( ~ S 1 , ~ S 2 ). We deﬁne them on product states via ~ S 1 ( | α i ⊗ | β i ) = ( ~ S | α i ) ⊗ | β i , and ~ S 2 ( | α i ⊗ | β i ) = | α i ⊗ ( ~ S | β i ) . Here the operators ~ S are the usual spin 1/2 operators (acting on a two-dimensional Hilbert space) that we have already discussed in some detail. If | α i is an eigenvector of spin along some axis, then so is | α i ⊗ | β i for any | β i . This means that if we know the spin component along the chosen axis with certainty for particle one then we get an eigenvector of the corresponding component of ~ S 1 , as we should. The same remarks apply to particle 2. The action of ~ S 1 and ~ S 2 are deﬁned on general vectors by expanding those vectors in a product basis, such as we considered above, and then using linearity to evaluate the operator term by term on each vector in the expansion. Sometimes one writes ~ S 1 = ~ S I, ~ S 2 = I ~ S to summarize the above deﬁnition. The two spin operators ~ S 1 and ~ S 2 commute (exercise) and have the same eigenvalues as their 1-particle counterparts (exercise). In this way we recover the usual properties of each particle, now viewed as subsystems. Total angular momentum There are other observables that can be deﬁned for the two particle system as a whole. Consider the total angular momentum ~ S , deﬁned by ~ S = ~ S 1 + ~ S 2 . You can easily check that this operator is Hermitian and that [ S k , S l ] = i ¯ klm S m , 1

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Physics 6210/Spring 2007/Lecture 26 so it does represent the angular momentum. Indeed, this operator generates rotations of the two particle system as a whole. The individual spin operators
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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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26 - Physics 6210/Spring 2007/Lecture 26 Lecture 26...

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