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I have a Hamiltonian of a semiclassical wavepacket in some external potential (say an EM field). enter image description here

I would like to linearize this Hamiltonian around a point $x_c$. Intuitively, I would simply try to do a "Taylor expansion," writing something like $H \approx H_c + (x-x_c)\frac{\partial H}{\partial x_c}$. However, the text I am following defines the first order term as:

enter image description here

My question is, how can I define the taylor series of a Hamiltonian to get this first order term, and how should I know that the series expansion needs both $(x-x_c)\frac{\partial H}{\partial x_c}$ as well as $\frac{\partial H}{\partial x_c}(x-x_c)$. It makes sense intuitively, but I am hoping for a slightly more formal reason and/or definition for a taylor series expansion of a Hamiltonian.

The paper I am following is: https://link.aps.org/doi/10.1103/PhysRevB.59.14915 In particular, I am looking at eq. 2.1 and 2.15.

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You need to add the other term so that the Hamiltonian is still hermitian. The adjoint of $$\hat{H}\approx\hat{H}(x_c)+(\hat{x}-x_c)\frac{\partial\hat{H}}{\partial \hat{x}}\Bigg|_{x_c}$$ is $$\hat{H}^\dagger=\hat{H}(x_c)^\dagger+\Bigg((\hat{x}-x_c)\frac{\partial\hat{H}}{\partial \hat{x}}\Bigg|_{x_c}\Bigg)^\dagger=\hat{H}(x_c)+\frac{\partial\hat{H}}{\partial \hat{x}}\Bigg|_{x_c}(\hat{x}-x_c)\neq \hat{H}.$$ However, you can see that if a symmetrization term is added, it becomes hermitian.

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