Answer the following questions: (1) Suppose that the annihilation and creation operators of the linear harmonic oscillator and 6, respectively, act on the eigen-state | n > and give the eigen- equations: |n>= |n 1 > and [n>= n + 1 + 1 >. Prove that: (i) 6|n>= n - 1 n - 2 -3> (iii) < n (b-b)|n>= -2n-1 (2) Suppose that the Hamiltonian operator for the linear harmonic oscillator is given by mw h where = Prove that: (1) b = (A + B) ( 2 - and B (iii) = -(A+ B)hw 2 2mw - = p 1 2m mwh + px. -mwx 2 (ii) < n | ( + )|n > = 2n + 1 (iv) [ t, t3 = 3 6t3 (ii) b (A - iB) (iv) = (bb + )w (3) Use the annihilation and creation operators of the linear harmonic oscillator ( and ) to calculate the matrices , , and . Note that: = (6 +6) and x = = hmw (b-6).