Finding the time evolution of a state

[Kyrios]

Homework Statement

1) Using energies and eigenstates that I've worked out, find time evolution ψ(t) of a state that has an initial condition ψ(0) = <br /> \begin{pmatrix}<br /> 1 \\<br /> 0\\<br /> \end{pmatrix}<br />

2) Find the expectation values < Sy> and <Sz> as a function of time.

Homework Equations

The Hamiltonian is H = \alpha (B_x S_x + B_y S_y + B_z S_z)


The energies that I worked out were the eigenvalues:

\lambda_1= \frac{ \alpha \hbar B_x }{2}

\lambda_2= - \frac{ \alpha \hbar B_x }{2}


The eigenstates were the eigenvectors
<br /> \begin{pmatrix}<br /> 1 \\<br /> 1\\<br /> \end{pmatrix}<br />

and

<br /> \begin{pmatrix}<br /> 1 \\<br /> -1\\<br /> \end{pmatrix}<br />

The Attempt at a Solution

I tried using the time evolution operator
U(t)= exp( \frac{ -i H t}{\hbar} )

I ended up with something that looks like this:

\psi(t) = A exp( \frac{ -i \alpha B_x t}{2} ) (1, 1) + B exp( \frac{ i \alpha B_x t}{2} ) (1, -1)


But I'm really unsure of where to go from here, or whether this is even right.

 

[Thoros]

Just check if your answer satisfies the inital condition and determine the constants A and B from normalization.