Question

H = omega sum_{i=1}^{3} (b_i b^i + frac{1}{2}) be the Hamiltonian of the quantum SHO, where the b^i, b_i are boson creation (resp annihilation) operators satisfying the commutation relations [b_i, b^j] = delta^j_i, [b_j, b_k] = [b^j, b^k] = 0. Show that: (i) The operators b^i_j = b^i b_j satisfy the gl(3, C) commutation relations [b^i_j, b^k_l] = delta^k_j b^i_l - delta^i_l b^k_j. (ii) The operators L_j = -iepsilon_{jkl} b^k_l (summation over repeated indices) satisfy the angular momentum commutation relations [L_j, L_k] = iepsilon_{jkl} L_l with epsilon_{ijk} the alternating tensor. (iii) The operators L_{pm} = L_1 pm iL_2, L_0 = L_3 satisfy the o(3, C) commutation relations of Q1.

          H = omega sum_{i=1}^{3} (b_i b^i + frac{1}{2})
be the Hamiltonian of the quantum SHO, where the b^i, b_i are boson creation (resp annihilation) operators satisfying the commutation relations
[b_i, b^j] = delta^j_i, [b_j, b_k] = [b^j, b^k] = 0.
Show that:
(i) The operators b^i_j = b^i b_j satisfy the gl(3, C) commutation relations
[b^i_j, b^k_l] = delta^k_j b^i_l - delta^i_l b^k_j.
(ii) The operators
L_j = -iepsilon_{jkl} b^k_l
(summation over repeated indices) satisfy the angular momentum commutation relations
[L_j, L_k] = iepsilon_{jkl} L_l
with epsilon_{ijk} the alternating tensor.
(iii) The operators
L_{pm} = L_1 pm iL_2, L_0 = L_3
satisfy the o(3, C) commutation relations of Q1.

$H = omega sumi=1^3 (bi b^i + frac12) be the Hamiltonian of the quantum SHO, where the b^i, bi are boson creation (resp annihilation) operators satisfying the commutation relations [bi, b^j] = delta^ji, [bj, bk] = [b^j, b^k] = 0. Show that: (i) The operators b^ij = b^i bj satisfy the gl(3, C) commutation relations [b^ij, b^kl] = delta^kj b^il - delta^il b^kj. (ii) The operators Lj = -iepsilonjkl b^kl (summation over repeated indices) satisfy the angular momentum commutation relations [Lj, Lk] = iepsilonjkl Ll with epsilonijk the alternating tensor. (iii) The operators Lpm = L1 pm iL2, L0 = L3 satisfy the o(3, C) commutation relations of Q1.$

Added by Brent M.

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