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Published on: June 8, 2018
Exactly and quasi-exactly solvable 'discrete' quantum mechanics
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan. ryu.yukawa.kyoto-u.ac.jp
Researchers introduce a straightforward method for creating exactly and quasi-exactly solvable (QES) Hamiltonians in discrete quantum mechanics. This approach utilizes the sinusoidal coordinate to reproduce known Hamiltonians and discover new ones.
Area of Science:
- Quantum Mechanics
- Mathematical Physics
Background:
- Discrete quantum mechanics offers a framework for studying quantum systems with discrete spectra.
- Exactly and quasi-exactly solvable (QES) systems are crucial for understanding quantum mechanics and finding analytical solutions.
Purpose of the Study:
- To present a simple recipe for constructing exactly and QES Hamiltonians in one-dimensional discrete quantum mechanics.
- To reproduce known Hamiltonians and discover new ones within this framework.
Main Methods:
- Utilizing intertwining relations, shape invariance, and Heisenberg operator solutions.
- Employing annihilation/creation operators and dynamical symmetry algebras, such as the q-oscillator and Askey-Wilson algebras.
- Leveraging the sinusoidal coordinate transformation.
Main Results:
- A recipe for constructing exactly and QES Hamiltonians in discrete quantum mechanics is successfully presented.
- The method reproduces all known Hamiltonians whose eigenfunctions are Askey scheme hypergeometric orthogonal polynomials.
- Several novel exactly and QES Hamiltonians have been constructed using this approach.
Conclusions:
- The sinusoidal coordinate is essential for constructing exactly and QES Hamiltonians in discrete quantum mechanics.
- The presented recipe offers a systematic way to generate solvable quantum models.
- This work expands the family of exactly and QES systems in quantum mechanics.
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