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Is there any relation between the automorphism of the group and group of permutation? If exists, give example.

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The three matrix operators for spin one satisfy sz Sy – Sy 8z = isz and cyclic permutations. Show that s = sz, (sz tisy )³ = 0. For the sane in, m, can have the values im, m – 1, .., -m, while A12 has eigenvalue m(m + 1). Thus M? = m(m + 1) ±2 5 x 1 = 5 times each once 6 15 4 x 8 = 32 times +3/2 ±1/2 3/2 each 8 times 4 ±1 3 x 27 = 81 times 1 each 27 times 3 2 x 18 = 96 times 1/2 ±1/2 each 48 times 1 × 12 = 42 times 0, each 42 times Total 256 eigenvalues A certain state | 4) is an eigenstate of L? and L,: L'|v) = 1(l+ 1) h² |»), mh|v) . For this state calculate (La) and (L²).
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Calculate the hermitian conjugate (adjoins) for operator d/dx
A spin-2 system has 5 spin states in the Sz basis. Express the following as matrices or vectors in this basis: Ŝ2 operator, Ŝx operator, Ŝy operator, Ŝz operator, and State |2, 1⟩x (x component of spin = ħ).