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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}

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

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The matrix exponential for any Pauli matrix using the general formula [see general formula][1]. This procedure gives,

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

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  • $\begingroup$ @Qmechanic, could you check my procedure to find the time evolution for the given Hamiltonian and the spinor wavefunction as the initial condition? $\endgroup$ – user0322 Apr 2 '15 at 21:29

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