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test1c - Physics 463 Test I Solve 4 of the following 6...

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Physics 463 - Test I: Solve 4 of the following 6 problems 1. Consider a quantum system in an eigenstate j ¿; j; m i .of J 2 and J z . (a) Compute for this state the mean values h J x i ; h J y i ; h J z i ; h J 2 x i ; h J 2 y i ; and h J 2 z i : (b) Use the results to compute the statistical uncertainties ¢ J x and ¢ J y associated with a measurement of a component of angular momentum perpendicular to the z axis.[Recall ¢ A = p h A 2 i ¡ h A i 2 ]. (c) If the system is in such an eigenstate of J 2 and J z ; is “the component of ~ J perpendicular to the z axis” a well-de…ned quanity? 2. The wave function of a particle is given by à ( ~ r ) = A h a 2 + ( x + iy ) 2 i e ¡ ar r 2 : where A and a are real constants. (a) Is à an eigenfunction of L 2 ? If so, what is the eigenvalue? If not, what values can be obtained if L 2 is measured? (b) Is à an eigenfunction of L z ? If so what is the eigenvalue? If not, what values can be obtained if L z is measured?
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