Question

(L43*) Spin can be represented by matrices. Show that all three spin matrices l 0 2 0 -1 0),"2=2 1 have eigenvalues of +1/2h and -1/2h. Calculate the corresponding eigenfunctions which we will denote as α-and β-eigenfunctions corresponding to spin l/2 particles. Show that Sj can be determined by the commutation of the other two matrices sn and sm, n, maj. Prove that the (2×2) matrix sz-s' +ss+s, commutes with all spin matrices, ie. s2s,-sis-. Calculate the eigenvalues of s2. The matrices si, s2 and s3 are called Pauli spin matrices or simply spin- operators. The operator s2 is the so-called Casimir invariant of the Lie-algebra of spin- matrices (see Exercise (2.4). Prove that any unitary transformation of si, s2 and s3 leads to the same set of eigenvalues and propose a different unitary transform of the Pauli-matrices. What happens to the corresponding eigenfunctions?

Answer 1

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