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Just part e
Transcribed Image Text: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).
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