QUESTION 2. Matrix quantum mechanics: The Hamiltonian of a two-state system is given by H = other operator by A = ( 1 -2 -2 1 ( hw 0 and an- 0 -ħw Hint: recall that expectation values of operators in matrix quantum mechanics are defined as (0) = COC where O is the matrix form of the operator and C is the state; either C(0) or C(t) in the examples below. (a) Write down the energy eigenvalues, and the energy eigenvectors U(1) and U(2) of H. (b) Show that V = ( is an eigenvector of A, by considering AV = XV. Find A and show that V is normalised correctly. (c) At t = 0 the system is in the state C(0) = V. Find the expectation values (H) and (H2). (d) Assume the system is in the state C(0) = V at t = 0. Using the expansion theorem find the state C(t) at later times >0. Hint: write V as a linear combination of U(1) and U(2) and put "wiggle factors". (e) Calculate the expectation values (A) and (A2) using the time dependent state C(t).

QUESTION 2. Matrix quantum mechanics: The Hamiltonian of a two-state system is given by H = other operator by A = ( 1 -2 -2 1 ( hw 0 and an- 0 -ħw Hint: recall that expectation values of operators in matrix quantum mechanics are defined as (0) = COC where O is the matrix form of the operator and C is the state; either C(0) or C(t) in the examples below. (a) Write down the energy eigenvalues, and the energy eigenvectors U(1) and U(2) of H. (b) Show that V = ( is an eigenvector of A, by considering AV = XV. Find A and show that V is normalised correctly. (c) At t = 0 the system is in the state C(0) = V. Find the expectation values (H) and (H2). (d) Assume the system is in the state C(0) = V at t = 0. Using the expansion theorem find the state C(t) at later times >0. Hint: write V as a linear combination of U(1) and U(2) and put "wiggle factors". (e) Calculate the expectation values (A) and (A2) using the time dependent state C(t).