wavefunction - How to find the time evolution for two-component spinor?

I would like to find the time evolution for the given Hamiltonian, the initial state of the system we choose two spinor wavefunction $\psi_{+}(t=0)$ and $\psi_{-}(t=0)$ as given below:

The effective Hamiltonian can be written as

$$\begin{equation} H=\nu_{F} {\bf \sigma}\cdot\left(q-By\hat x\right) \end{equation}$$ where ${\bf \sigma}=(\sigma_{x},\sigma_{y})$ and $q=(q_{x},q_{y})$ are the Pauli matrices and the momentum operator respectively. Taking the spinor wavefunctions as $$\begin{align} \psi_{+}&=e^{ikx}\\ \psi_{-}&=e^{ikx} \label{eq:wavefs} \end{align}$$ as the initial condition for the Schrodinger equation. The time evolution of the system, $$\begin{equation} \Psi(r,t)=e^{-i\nu_{F} {\bf \sigma}\cdot\left(q-By\hat x\right)t}\Psi(r,0) \end{equation}$$ and $$\begin{equation} \Psi(r,0)=e^{ikx}\begin{pmatrix} 1\\ 1 \end{pmatrix} \end{equation}$$