Quantum mechanics-hamiltonian matrix and stationary states

mimocs

Homework Statement

let [1> and [2> mutually orthogonal states (eigenstates of some Hermitian operator).
the Hamiltonian operator is given by H=c[1><2]+c[2><1], where c is a real number.

(a) calculate the eigenstates and corresponding eigenvalues of H
(b) if the initial state of the system is [1>, how does the state evolve with time?

Homework Equations

H[ψ>=ih$\frac{∂[ψ>}{∂t}$

The Attempt at a Solution

for (a), I first earned hamiltonian matrix as below
0 c
c 0

from this, I earned two eigenvalues +c and -c.
and earned $\sqrt{1/2}$( 1 1 ) and $\sqrt{1/2}$( 1 -1 ) as two eigenstates of hamiltonian matrix.

however what I don't know is the answer of (b)
the problem I was given does not have any information of the state [1>. Is there anyway to solve this problem? if I assume [1> as (1 0) (and [2> as (0 1)) this problem can be solved. but I don't know more... can anyone help me?