# Using the time evolution operator

1. Jun 1, 2010

### ian2012

I hope someone can help me out here,

I am confused with a line of text I read - it is an example of a 2D Hilbert space with orthonormal basis e1, e2. The Hamiltonian of the system is the Pauli matrix in the y-direction. Given by the matrix:

$$\sigma_{y} = (\frac{0, -i}{i, 0})$$

The eigenvectors of the Hamiltonian are given by:

$$| \pm >_{y}= \frac{1}{\sqrt{2}}(| e_{1} > \pm i|e_{2}>)$$

So, applying the time evolution operator to the eigenvectors gives:

$$U| \pm >_{y}=exp(\frac{-i(t-t_{0}) \sigma_{y}}{\hbar})| \pm >_{y}$$
$$U| \pm >_{y}=exp(\frac{\mp i(t-t_{0})}{\hbar})| \pm >_{y}$$

I don't understand how the last line came about?

2. Jun 1, 2010

### bapowell

The last line follows because the eigenvalues of $$\sigma_y$$ are +/- 1.

3. Jun 1, 2010

### ian2012

Oh right, of course, so it let's you simplify the expression.