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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online