# Numerical approximation to eigenstates and their differentials

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).

• 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. – AHusain Dec 29 '18 at 20:42