3 1 H = ? ? (b?b?? + (1)/(2)) i=1 be the Hamiltonian of the quantum SHO, where the b?, b? are boson creation (resp annihilation) operators satisfying the commutation relations [b?, b??] = ???, [b?, b?] = [b??, b??] = 0. Show that: (i) The operators b?? = b??b? satisfy the gl(3, C) commutation relations [b??, b??] = ???b?? - ???b??. (ii) The operators L? = -i????b??b? (summation over repeated indices) satisfy the angular momentum commutation relations [L?, L?] = i????L? with ???? the alternating tensor. (iii) The operators L? = L? ± iL?, L? = L? satisfy the o(3, C) commutation relations of Q1.

3 1 H = ? ? (b?b?? + \frac{1}{2}) i=1 be the Hamiltonian of the quantum SHO, where the b?, b? are boson creation (resp annihilation) operators satisfying the commutation relations [b?, b??] = ???, [b?, b?] = [b??, b??] = 0. Show that: (i) The operators b?? = b??b? satisfy the gl(3, C) commutation relations [b??, b??] = ???b?? - ???b??. (ii) The operators L? = -i????b??b? (summation over repeated indices) satisfy the angular momentum commutation relations [L?, L?] = i????L? with ???? the alternating tensor. (iii) The operators L? = L? ± iL?, L? = L? satisfy the o(3, C) commutation relations of Q1.

If the general angular momentum quantum number j is 1, there is a triplet of |j,mj) states: {1,1), |1,0), and |1,-1). In this case, a matrix representation for the operators Jz, Jy, and J can be constructed if we represent the |j,mj) triplet by three-component column vectors as follows: |1,1) [1,0) (8), |1,-1) -(8) (4.1). J can then be represented by the matrix: J = h Construct matrix representations for the raising and lowering operators, J+ and J_, acting on the eigenstates |1,1), |1,0), and |1,-1) in the representation given in equation (4.1). Use the relationships J^2 = (J+J-) + Jz Jy = (J- - J+)/2 to construct matrix representations for J and Jy. Show that the matrix representations of Jz, Jy, and J obey the commutation relation [Jz, Jy] = ihJ.

Zhaojie X.

Question

Instant Answer