The three matrix operators for spin one satisfy sz Sy – Sy 8z = is, and cyclic permutations. Show that s = sz, (sz tisy)3 = 0. For the same in, m, can have the valucs im, m – 1, .., -m, while A12 has eigenvalue m(m + 1). Thus M2 = m(m + 1) +2 土1 each once 6 5 x1 = 5 times 15 +3/2 ±1/2 3/2 each 8 times 4 x 8 = 32 times 4 1 each 27 times 3 x 27 = 81 times 3 1/2 ±1/2 each 48 times 2 x 18 = 96 times 0, each 42 timcs 1x 12 = 42 times Total 256 eigenvalues A certain state | ) is an eigenstate of L2 and L,: L'|v) = 1(l+ 1) h² |), mh| v). For this state calculate (La) and (L2).

results. OAKL Consider a pne dimensional infinite-wall potential V = ∞ for > Land z<0, and V0 for There are two identical spin half fermions in this potential well and spin states are dened by) and 4) for spin up and down, respectively. a) (Very easy) Find the eigenvalues and the corresponding wave functions for the potential well. b) (Ensy) Write the full wave function for the fermions in singlet state with possible minimum energy. e) (Easy) Write the full wave function for the fermions in triplet state with possible minimum energy

Please hand writing answer.

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).