11 - Physics 6210/Spring 2007/Lecture 11 Lecture 11...

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Physics 6210/Spring 2007/Lecture 11 Lecture 11 Relevant sections in text: § 1.7, 2.1 Product observables We have seen how to build the Hilbert space for a composite system via the tensor product construction. Let us now see how to build the observables. Let A 1 be an observable for system 1 and B 2 an observable for system 2. We define the corresponding observables for the composite system as follows. Consider the following linear operators on product vectors: A 1 I | α i ⊗ | β i = ( A 1 | α i ) ⊗ | β i , I B 2 | α i ⊗ | β i = | α i ⊗ ( B 2 | β i ) . We extend the definitions of these operators to general states by expanding the state in a product basis and demanding linearity of the operator. Thus defined, these operators are Hermitian. Moreover, they commute (good exercises!): [ A 1 I, I B 2 ] = 0 , which means that one can still determine with statistical certainty the subsystem observ- ables within the composite system. For example, if we are looking at 2 particles, it is possible to ascertain both particle 1’s position and particle 2’s momentum with arbitrarily good statistical accuracy. Likewise, when using this contruction to build a model for a particle moving in 3-d from our 1-d model, we end up with all the position variables com- muting among themselves and all the momentum variables commuting among themselves. Of course, the commutation relations obtained in the subsystems appear in the combined system. Exercise Show that if [ A 1 , B 1 ] = C 1 then [ A 1 I, B 1 I ] = C 1 × I, thus explaining the generalization of the canonical commutation relations we obtained for a particle in 3-d. Please take note that it is conventional in many references to simply drop the I or I factors when denoting the extension of the subsystem operators to the composite system. Indeed, it is common to also drop the symbol entirely! Also note that not every observable of the system is of the above form. As usual, any self-adjoint operator can represent an observable. Thus there will be observables that refer 1
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Physics 6210/Spring 2007/Lecture 11 to the composite system as a whole ( e.g., the total energy) and are not observables of either of the subsystems alone. Returning to our example of a particle moving in 2-d. We have position operators for
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11 - Physics 6210/Spring 2007/Lecture 11 Lecture 11...

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