# Finding the time evolution of a state

1. Mar 10, 2013

### Kyrios

1. The problem statement, all variables and given/known data

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.

2. Relevant 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

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

3. 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.

Last edited: Mar 10, 2013
2. Mar 10, 2013

### Thoros

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