Question

Problem 2 Ul Consider twovectors, v and u , where Vj,Uj are complex U2 numbers a. Find the conditions that ensure normalization for each of these vectors b. Write down explicitly the tensor product v&u as a four-component vector c. Consider a square matrix A acting on v and a square matrix B acting on u, show that (AS>B) (v u)-Au Bu Using Dirac notation for the vectors: v- |v), u-|u) d. Write down the normalization condition for each vector e. Express |v) and lu) in terms of basis vectors l0) and |1) and repeat a f. Introduce a second vector for each space, v' and u' and write down the condition for each of the v, v' and u, u' to form an orthonormal set g. Write down symbolically the tensor product of |u) and lv) in two equivalent wavs

Answer 1

SOLUTION

A,B

normalization of each vector, v = 1/squareroot v_1^2 + v_2^2 (v_1 v_2) u = 1/squareroot u_1^2 + u_2^2 (u_1 u_2) tensor product v times u = (v_1 v_2) times (u_1 u_2) = (v_1 u_1 v_1 u_2 v_2 u_1 v_2 u_2) A v = (A_1 A_3 A_2 A_4) (v_1 v_2) = (A_1 v_1 + A_2 v_2 A_3 v_1 + A_4 V_4) B u = (B_1 B_3 B_2 B_4) (u_1 u_2) = (B_1 u_1 + B_2 u_2 B_3 u_1 + B_4 u_2) A v times B u = ((A_1 V_1 + A_2 v_2) (B_1 u_1 + B_2 u_2) (A_1 v_1 + A_2 v_2) (B_3 u_1 + B_4 u_2) (A_3 v_1 + A_4 V_2) (B_1 u_1 + B_2 u_2) (A_3 v_1 + A_4 v_2)

C