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1.The Hamiltonian of a two-state system is:

Ap = (2 721 ²-21)

a) (1 point) Verify that He is Hermitian.

b) (3 points) Find eigenvalues of Fo

c) (3 points) Find eigenvectors of Ho

d) (3 points) Consider the state ) = (

What are the possible outcomes of the experiment in which the energy of the particle in state

1o) is measured? What are the probabilities of the possible outcomes?

Hint: It would be easier if you normalize (o) and the two eigenvectors first.

e) (2 points) Solve the time-dependent Schrödinger equation:

dly)

dt

ih

-=Foly)

with the initial condition: (t = 0)) = 1o) = (3), i.e., derive expressions for the components

a(1) and b() of the solution 14(t)) = (())

Fig: 1