2. Uncertainty and uncertainty relation for a spin-1/2 system. (a) Use the matrix representation of each component S, of

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2 Uncertainty And Uncertainty Relation For A Spin 1 2 System A Use The Matrix Representation Of Each Component S Of 1 (38.2 KiB) Viewed 31 times

2. Uncertainty and uncertainty relation for a spin-1/2 system. (a) Use the matrix representation of each component S, of S given by S₁ = ₁, i = 1, 2, 3 to verify the commutation relations given by [Sj, Sk] = ihejki St. Here Ejkt is the totally antisymmetric tensor of rank three, defined as for an even permutation of 1,2,3 -& -1 for an odd permutation of 1,2,3 otherwise (b) A spin-1/2 particle is in the state given by ly) = c[(5 + 6i)|+)+(2-3i)|-)] where c is a real normalization factor. i. Find c. Ejkl ii. Determine (Sx), (S2), (Sy), (S3), (S₂), (S2), ASx, AS, and ASz. iii. Use the uncertainty relation to obtain the lower bounds of the uncertainty products AS AS, AS AS₂, and AS,AS, and compare with the corresponding exact uncertainty products calculated using the exact uncertainties calculated in b.