Quantum Particle on Dual Weight Lattice in Weyl Alcove
Abstract
:1. Introduction
2. Dual-Weight Fourier–Weyl Transforms
2.1. Weyl Groups and Invariant Shifted Lattices
2.2. Signed Fundamental Domains
2.3. Signed Dual Fundamental Domains
2.4. Dual-Weight Discretization of Weyl Orbit Functions
3. Dual Weight Lattice Models
3.1. Dual-Weight Dots
3.2. Schrödinger Equations
3.3. Time Evolution
4. Dual Weight Lattice Models of and
4.1. Case
4.2. Case
5. Conclusions
- The developed one-particle dual-weight discrete quantum billiard systems describe the non-relativistic quantum particle propagating on the dots , which comprise finitely-many positions located inside the scaled closure of the Weyl alcove . The precise arrangements (15) and (16) of the Dirichlet and Neumann walls and that realise the quantum trapping of the particle and coincide with the dual-root billiards [1] constitute the boundaries of the simplex . Any predetermined admissible dual-weight hopping function , which encodes the amplitude propagation to the neighbouring positions, directly provides explicit formulas for the eigenenergies of the systems (59) via its Fourier transform by the symmetric Weyl orbit sums over its finite dominant support . The vectors of the orthonormal momentum basis, , , determined independently on the dual-weight hopping function by their explicit form (56) constitute solutions of the time-independent Schrödinger Equation (58). The time evolution of the dual-weight quantum systems from any normalised initial state vector given in the position basis |a〉, is exactly determined (71).
- The presence of the affine Weyl group orbits of the target positions in the coupling sets (51) represent the first essential symmetry component for implementing the interactions enforced by the boundaries of . Secondly, the addition of the sign -function (13) values over the affine-reflected positions in the coupling set counts the number and type of amplitude reflections between the source position and the target position . The -function generalises the sign functions from [40] that are necessary for describing the Galois symmetries of Weyl orbit functions. The square roots of the stabiliser ε-functions (7) present as factors in defining relation of the dual-weight hopping operators matrix elements in the position basis (52) and manifest a direct consequence of the weighted discrete orthogonality relations (39). The ε-function subsequently straightforwardly regulates the probabilities (74) of finding the particle in a stationary state on the Neumann walls of the simplex . The Neumann boundary effect, which is observed similarly in dual-root models [1], is pointedly evident in Figure 3 and Figure 6.
- Considering an electron as the quantum particle propagating in the current discrete quantum systems, a novel class of the tight-binding models [3] with the electron propagating among atoms positioned at the points of the dual-weight dot is obtained. The hopping integrals [15] between the coupled neighbouring positions in the atomic lattice, which might be estimated from theoretical considerations and/or fine-tuned experimentally, directly enter the present models as the values of the dual-weight hopping function . Analogously to the dual-root models, the physical interpretation of the dual-weight models coincides with the inductively developed electron propagation in a crystal lattice [41]. Since the dual-weight Fourier–Weyl transforms of the current one-dimensional model of a linear crystal specialise to the four types I–IV of the discrete cosine and sine transforms [18,25], the current stationary state vectors represent boundary-dependent (anti)symmetric alternatives to the periodic exponential solutions [41]. Moreover, the discrete Hamiltonian approach used for dual-weight and dual-root models produces strictly defined boundary-dependent forms of the energy spectra (59).
- Similarly to the dual-root models, the dual-weight models employ the generalised dual-weight Fourier–Weyl transforms (41) to construct the momentum basis (56) together with the stationary states (70) and time-evolutions (71). The utilisation of the weight lattice transforms [23] as well as the dual-weight E-transforms [42] for the description of analogous discrete quantum systems deserves further study. Potentially resulting in the generalisation of the present models to the prominent honeycomb-type (pseudo)lattices [17,32], the intricate composition of the weight and root lattice transforms demands a specific construction of extension coefficients of the extended Weyl orbit functions [43]. The calculation of the extension coefficients is determined by the desired form of product-to-sum decomposition formulas (33), which characterise the coupling of the considered (pseudo)lattice model. Since the extended Weyl orbit function approach potentially represents alternative description to the (pseudo)spinor wavefunctions approach [17], the Fourier–Weyl transforms induced by the extended Weyl orbit functions, together with the discrete symmetry analysis of the associated quantum systems, deserve further study.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Brus, A.; Hrivnák, J.; Motlochová, L. Quantum Particle on Dual Weight Lattice in Weyl Alcove. Symmetry 2021, 13, 1338. https://doi.org/10.3390/sym13081338
Brus A, Hrivnák J, Motlochová L. Quantum Particle on Dual Weight Lattice in Weyl Alcove. Symmetry. 2021; 13(8):1338. https://doi.org/10.3390/sym13081338
Chicago/Turabian StyleBrus, Adam, Jiří Hrivnák, and Lenka Motlochová. 2021. "Quantum Particle on Dual Weight Lattice in Weyl Alcove" Symmetry 13, no. 8: 1338. https://doi.org/10.3390/sym13081338
APA StyleBrus, A., Hrivnák, J., & Motlochová, L. (2021). Quantum Particle on Dual Weight Lattice in Weyl Alcove. Symmetry, 13(8), 1338. https://doi.org/10.3390/sym13081338