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Quantum Mechanics with Matrices (2020 - 2021)
Matrix mechanics is a formulation stemmed from quantum mechanics proposed by Heisenberg. It provides an enlightened method to represent various quantum properties as a matrix. In this research, we studied the pentadiagonal matrix with two coupling parameters, v and w. In the case of w = 0, which we currently considered, it turns out to be a tridiagonal matrix. In our model, all values on the below and above diagonals are the same, that is, v. We described the matrices with energies nE (with n = 0, 1, …,10) along the diagonal and the one with all cases of E = 0 along the diagonal.
By listing the resultant wavefunctions in both cases, we noticed the symmetric patterns emerged. For the energies nE’s case under the strong coupling regime, we found that the wave functions strikingly fall into 2 classes and we could assign a quantum number to distinguish them. Furthermore, we employed the perturbation theory into our assumptions and checked if the results are consistent.
We also demonstrated the transition from the rotational spectrum to the vibrational spectrum. Also, we found more symmetric patterns when varying different values of v and w. Another result about the asymptotic behavior as zero-diagonal matrix under strong coupling is also an undergoing research.
Relevant resources:
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C. Fan and L. Zamick (2021). Matrix Model: Emergence of a Quantum Number in the Strong Coupling Regime. DOI: 10.13140/RG.2.2.24099.37921 [PDF]
This work is supported by Aresty Research Center and has been presented in the 17th Aresty Annual Symposium. The poster is shown here (click and enlarge the poster):
Please click here to watch the virtual symposium video (3 minutes only).