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03252010b - Quantum Mechanics Commutation 5 april 2010 I...

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Quantum Mechanics: Commutation 5 april 2010 I. Commutators: Measuring Several Properties Simultaneously In classical mechanics, once we determine the dynamical state of a system, we can simultaneously obtain many different system properties (i.e., ve- locity, position, momentum, acceleration, angular/linear momentum, kinetic and potential energies, etc.). The uncertainty is governed by the resolution and precision of the instruments at our disposal. In quantum mechanics, the situation is different. Consider the following: 1. We would like to measure several properties of a particle repre- sented by a wavefunction. 2. Properties of a q.m. system can be measured experimentally. Theo- retically, the measurement process corresponds to an operator acting on the wavefunction. The outcomes of the measurement are the eigenvalues that correspond to the operator. The operator is taken to be acting on a wavefunction that is either a pure eigenfunction of the operator of interest, or an expansion in the basis of eigenfunctions. In order to measure, for instance, 2 properties simultaneously , the wave-
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