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I am working in Adiabatic Quantum Computing and I have a $6\times6$ Hamiltonian. I have only the symbolic expression for its eigenstates which have complicated expressions in solutions of degree $6$ characteristic polynomial of the Hamiltonian and even ordinary computing over these eigenstates is beyond the power of Mathematica in my computer.
I need to evaluate the Geometric phases terms like $\langle\psi_n|\partial _{\phi}|\psi_n\rangle$ where $\phi$ is just a driving adiabatic parameter and $|\psi_n\rangle$ are the solutions to the Hamiltonian.

Now, I can get numerics for $|\psi_n\rangle$ but once I get that, how can I evaluate the differential $\partial_{\phi}|\psi_n\rangle$? Or to even approximate it in any form? Because it is impossible to get a manipulative symbolic form for $|\psi_n\rangle$ with such messy calculations, all that can be done is to write Hamiltonian numerically and find the obvious numerical eigenstates.

I have tried using all kinds of Numerical functions available in Mathematica but such a numerical differentiation in one parameter is not possible for this kind of complexity (takes a lot of time).

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    $\begingroup$ There is also mathematica.stackexchange.com if you are interested in how to write faster numerics in Mathematica. Like by modifying the precision you demand. $\endgroup$ – AHusain Dec 29 '18 at 20:42

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