13-5 - Physics 6210/Spring 2007/Lecture 6 Lecture 6...

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Unformatted text preview: Physics 6210/Spring 2007/Lecture 6 Lecture 6 Relevant sections in text: 1.4 Complete set of commuting observables Only in the simplest physical systems will the measurement of a single observable suffice to determine the state vector of the system. Of course, the spin 1/2 system has this property: if you measure any component of the spin the state vector is fixed (up to an irrelevant phase factor*). More complicated systems need more observables to char- acterize the state. For example, recall that the energy levels of a hydrogen atom can be uniquely specified by the energy, orbital angular momentum magnitude, one component of the angular momentum and the electron spin state. Other choices are possible. After measuring an observable and getting a particular outcome an eigenvalue the state of the system is the corresponding eigenvector. It may happen that more than one state vector is associated with that measurement outcome. Mathematically, the eigenvalue is degenerate. To pin down the state uniquely, other observables must be measured. Evi- dently, if we have a set of observables whose measurement uniquely determines the state (vector up to a phase), then that state vector must be a simultaneous eigenvector of all the observables in question. If we demand that the possible outcomes of measurements of this set of observables defines a basis for the Hilbert space, then the observables must be compatible, i.e., their operator represntatives must all commute. Such a set of variablestheir operator represntatives must all commute....
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13-5 - Physics 6210/Spring 2007/Lecture 6 Lecture 6...

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