1. Compute the eigenvalues and eigenvectors of al, a2 and a3 where  

cri = (0 1) ‘ 0.2 = 0—i (1 0 ) 0-3 = 1 0 i 0 0 —1  

are the Pauli spin matrices. Normalize each eigenvector 1v) so that (v Iv) = 1.  

(1)  

2. Let I1),12),13) be an orthonormal basis for a three dimensional vector space V. Find the eigenvalues and eigenvectors of the operator  

A= (11) +12) +13))((11+ (21+ (31)  

(2)  

Make sure to choose the eigenvectors so that they are orthonormal. 3. The exponential of a matrix A is defined as eA = E,7_0 An;. (a) Compute ei’a3 where cr3 is the Pauli matrix defined above, and 0 is a scalar. (b) Compute ei°’. 4. Show that (vIA210 is real and greater than or equal to 0 for any Hermitian operator A and any vector v.  

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