[Physics] Unitary transformation of the Hamiltonian with spin-orbital coupling

hamiltonianhomework-and-exercisesquantum mechanics

I am reading this Paper recently. The author says that:
for this Hamiltonian: $$H(t) = \frac{p^2}{2m} + \frac{m\omega^2}{2}x^2 + \alpha p_x \sigma_y$$

If we make a unitary transformation $\mathcal{A_{\alpha}} = e^{-imx\alpha \sigma_y/\hbar}$, The Hamiltonian will be transformed to
$$
H_0 = \frac{p^2}{2m} + \frac{m\omega^2}{2}x^2
$$
And after that, we can solve the Schrodinger Equation and the evolution of the states can be calculated.

I cannot figure out how to do the transform. Is it just $\mathcal{A_{\alpha}}H(t)\mathcal{A_{\alpha}}^{\dagger}$? (I failed while trying to calculate this).
Does anybody knows how to do the transformation?

Best Answer

For the right A take the CC of the exponent and for the Pauli spin matrix take adjoint. Hope it works.

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