GenUncertPrinciple (1)

GenUncertPrinciple (1) - GeneralUncertaintyPrinciple...

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General Uncertainty Principle Michael Fowler 12/ 9/06 Uncertainty and Non­Commutation As we discussed in the Linear Algebra lecture, if two physical variables correspond to commuting Hermitian operators, they can be diagonalized simultaneously—that is, they have a common set of eigenstates. In these eigenstates both variables have precise values at the same time, there is no “Uncertainty Principle” requiring that as we know one of them more accurately, we increasingly lose track of the other. For example, the energy and momentum of a free particle can both be specified exactly. More interesting examples will appear in the sections on angular momentum and spin. But if two operators do not commute, in general one cannot specify both values precisely. Of course, such operators could still have some common eigenvectors, but the interesting case arises in attempting to measure A and B simultaneously for a state y in which the commutator [ ] , A B has a nonzero expectation value, [ ] , 0 A B . A Quantitative Measure of “Uncertainty” Our task here is to give a quantitative analysis of how accurately noncommuting variables can be measured together. We found earlier using a semi­quantitative argument that for a free particle, p x D &D :h at best. To improve on that result, we need to be precise about the uncertainty
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GenUncertPrinciple (1) - GeneralUncertaintyPrinciple...

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