Question

Suppose we have a quantum system with N eigenstates. Then we know the eigenstates can be expressed as vectors, and operators can be represented by N × N matrices (a) Prove that (A)(A)where At is the transpose conjugate of matrix A. Here, A is not required to be Hermitian operator (Hint: express A and) in matrix and vector form. Use matrix calculation to show that (Αψ|U) is the same as 1Atlp.) (b) Prove that (ΑΒψ|U)-(ψ1BtAtlp). Á and B are not required to be Her- mitian operators Hint: this is equivalent of proving matrix (AB)tBA, To prove two matrices equal to each other, you only need to show the i-th row, j-th column element of (AB)t is the same as the i-th row, jth column element of BtAt, (c) Proof that for Hermitian operator A, we have At definition of Hermitian operator eq (1) A. Hint: use the (d) If A and B are both Hermitian operators (matrices), is AB Hermitian op- erator? If yes, prove it. If no, show an example (e) Are Pauli matrices σι, Ơy, and σ2 Hermitian operators? Is σ1Oy Hmeritian operator? We have:

Answer 1

Take A = (A_11 A_12 - - - - - - A_1 N A_21 A_22 - - - - - - A_2 N - - - - A_N1 - - - - - - A_N N) < A psi Vertical-bar psi > take Vertical-bar psi > = [psi_1 psi_2 - - - psi_n] = (psi_1 A_11 + psi_2 A_12 + - - - - psi_N A_1N - - - psi_1 A_N1 + psi_2 A_N2 + - - - - + psi_N A_NN) (psi_1 - - - psi_N) = psi_1 A_11 psi_1 + psi_2 A_12 psi_1 + - - - + psi_N A_1 N psi_1 + - - - - + psi_1 A_N1 psi_N + - - - + psi_N A_NN psi_N < psi Vertical-bar A^+ Vertical-bar 4 > = (psi_1 - - - psi_N) (A_11 A_21 - - - A_N1 A_12 A_22 - - - A_N2 - - - - - - A_1N A_2N - - - A_NN) (psi_1 psi_2 - - - psi_n) = (psi_1 A_11