Numerical Calculation of Hamiltonian Overlaps

In summary, the speaker is reading a paper and wants to reproduce its results for N=5. They are struggling with a computationally taxing step and are asking for advice on how to deal with it numerically.
  • #1
thatboi
121
18
Hi all,
I am currently reading through this paper: https://iopscience.iop.org/article/10.1088/1367-2630/10/4/045030
and would like to reproduce their results for N=5.
My roadblock is with (9), which models the classical motion of the system. Now symbolically finding the eigenstates of the matrix (4) for N=5 is too computationally taxing to make this a viable choice. It would be easy to numerically calculate the overlap on its own, but my problem is with the gradient term. How should I deal with it numerically?
Thanks.
 
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  • #2
thatboi said:
It would be easy to numerically calculate the overlap on its own, but my problem is with the gradient term. How should I deal with it numerically?
It can be approximated using a finite difference method.
 

1. What is the Hamiltonian overlap?

The Hamiltonian overlap is a measure of the similarity between two quantum states, represented by their corresponding Hamiltonian matrices. It is used in many numerical calculations in quantum mechanics, such as in the calculation of transition probabilities and energy levels.

2. Why is it important to calculate Hamiltonian overlaps?

Calculating Hamiltonian overlaps is important because it allows us to compare the properties of different quantum states and understand their relationships. This is crucial in many applications, such as in designing quantum algorithms and predicting the behavior of complex quantum systems.

3. How is the Hamiltonian overlap calculated?

The Hamiltonian overlap is calculated by taking the inner product of the two Hamiltonian matrices, which involves multiplying the corresponding elements and summing them up. This can be done using numerical methods, such as matrix multiplication and integration.

4. What are some challenges in numerical calculation of Hamiltonian overlaps?

One of the main challenges in numerical calculation of Hamiltonian overlaps is the large size of the matrices involved, which can lead to long computation times and potential errors. Additionally, the accuracy of the results can be affected by the choice of numerical methods and the precision of the calculations.

5. How can the accuracy of Hamiltonian overlaps be improved?

To improve the accuracy of Hamiltonian overlaps, it is important to use efficient numerical methods and increase the precision of the calculations. Additionally, it can be helpful to use techniques such as matrix diagonalization and adaptive integration to reduce errors and improve the overall accuracy of the results.

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