# How to find the time evolution for two-component spinor? [closed]

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 $$H=\nu_{F} {\bf \sigma}\cdot\left(q-By\hat x\right)$$ 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, $$\Psi(r,t)=e^{-i\nu_{F} {\bf \sigma}\cdot\left(q-By\hat x\right)t}\Psi(r,0)$$ and $$\Psi(r,0)=e^{ikx}\begin{pmatrix} 1\\ 1 \end{pmatrix}$$

## closed as off-topic by ACuriousMind♦, Kyle Kanos, JamalS, Constandinos Damalas, Ryan UngerApr 9 '15 at 21:12

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. We want our questions to be useful to the broader community, and to future users. See our meta site for more guidance on how to edit your question to make it better" – ACuriousMind, Kyle Kanos, JamalS, Constandinos Damalas, Ryan Unger
If this question can be reworded to fit the rules in the help center, please edit the question.

$$\exp[-iHt] = I_{2\times2}\cos \nu_{F}t - i {\bf\sigma}\cdot\left(q-By\hat x\right) \sin \nu_{F}t$$ I tryed to solve using this method but seems not convincing. Any comments would be appreciated.