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) = $$\begin{pmatrix} 1 \\ 0\\ \end{pmatrix}$$

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
$$\begin{pmatrix} 1 \\ 1\\ \end{pmatrix}$$

and

$$\begin{pmatrix} 1 \\ -1\\ \end{pmatrix}$$

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.