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MÉTODOS ESPECTRALES SOBRE EL ÁTOMO DE HIDRÓGENO
MÉTODOS ESPECTRAIS NO ÁTOMO DE HIDROGÊNI
SPECTRAL METHODS ON THE HYDROGEN ATOM
Citacion sugerida: Reinoso Reinoso,
C., Abancin Ospina, R. (2024). Spectral
methods on the hydrogen atom.
Revista Bases de la Ciencia, 9(1), 15-
28. DOI: https://doi.org/10.33936/
revbasdelaciencia.v9i1.6403
Autor
Editor Académico
Recibido: 24/01/2024
Aceptado: 23/04/2024
Publicado: 24/04/2024
1
Escuela Superior Politécnica de
Chimborazo (ESPOCH), Riobamba,
Ecuador
* Autor para correspondencia.
Resumen
Este estudio investiga la implementación computacional de la teoría espectral y sus
aplicaciones en el análisis de problemas matemáticos complejos. Explora el uso de
lenguajes de programación modernos y bibliotecas cientícas para implementar
y visualizar conceptos matemáticos complejos, centrándose particularmente en
la teoría espectral dentro de la física cuántica. La investigación emplea métodos
computacionales para abordar los desafíos en la interpretación de propiedades
espectrales, utilizando el método espectral de Lanczos para el cálculo de valores
propios en matrices grandes y dispersas. Los resultados ilustran la ecacia de estas
técnicas computacionales para visualizar estados cuánticos, lo que demuestra el
potencial de la programación avanzada para comprender y resolver problemas
complejos en física cuántica y teoría de grafos espectrales. Los hallazgos del
estudio son importantes para unir los métodos computacionales con el análisis
espectral teórico, ofreciendo una nueva perspectiva sobre la aplicación de técnicas
computacionales en la investigación cientíca.
Palabras clave: Teoría espectral, Matemática computacional, Física cuántica,
Método Lanczos, Cálculos de valores propios.
Abstract
This study investigates the computational implementation of spectral theory and
its applications in analyzing complexmathematical problems. It explores the use
of modern programming languages and scientic libraries for implementing and
visualizing complex mathematical concepts, particularly focusing on spectral
theory within quantum physics. The research employs computational methods to
address the challenges in interpreting spectral properties, utilizing the Lanczos
spectral method for eigenvalue calculation in large, sparse matrices. The results
illustrate the eectiveness of these computational techniques in visualizing
quantum states, demonstrating the potential of advanced programming in
understanding and solving intricate problems in quantum physics and spectral
graph theory. The study’s ndings are signicantin bridging computational
methods with theoretical spectral analysis, oering a new perspective on the
application of computational techniques in scientic research.
Keywords: Spectral theory, Computational mathematics, Quantum physics,
Lanczos method, Eigenvalue calculations.
Resumo
Este estudo investiga a implementação computacional da teoria espectral e suas
aplicações na análise de problemas matemáticos complexos. Explora o uso de
linguagens de programação modernas e bibliotecas cientícas para implementar
e visualizar conceitos matemáticos complexos, concentrando-se particularmente
na teoria espectral dentro da física quântica. A pesquisa emprega métodos
computacionais para enfrentar os desaos na interpretação de propriedades
espectrais, utilizando o método espectral de Lanczos para cálculo de autovalores
em matrizes grandes e esparsas. Os resultados ilustram a ecácia dessas técnicas
computacionais na visualização de estados quânticos, demonstrando o potencial
da programação avançada na compreensão e resolução de problemas complexos
em física quântica e teoria de grafos espectrais. As descobertas do estudo são
signicativas ao unir métodos computacionais com análise espectral teórica,
oferecendo uma nova perspectiva sobre a aplicação de técnicas computacionais
na pesquisa cientíca..
Palavras chave: Teoria espectral, Matemática computacional, Física quântica,
Método Lanczos, Cálculos de valores.
César Daniel Reinoso Reinoso
1
Ramón Antonio Abancin Ospina
1,*
Oswaldo José Larreal Barreto
iD
iD
iD
15
Ciencias Matemáticas
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1. Introduction
The study of the hydrogen atom has been a essential pillar in theoretical physics and chemistry, oering a window into the
principles of quantum mechanics and fundamental matter interactions. Precision in the spectral analysis of the hydrogen
atom is crucial not only for theoretical understanding but also for practical applications in spectroscopy and quantum
technologies. Advances in computational tools, particularly with the use of Python, oer new possibilities to approach this
classic study with renewed and more precise methods.
Despite advances in computational techniques, implementing spectral methods in the study of the hydrogen atom faces
signicant challenges. These include the need for precise wave function discretization, ecient management of large
matrices in the Hamiltonian representation, and the ecient and accurate calculation of eigenvalues, which are crucial for
understanding the spectral properties of the atom.
The spectral analysis of quantum systems, particularly the hydrogen atom, remains a cornerstone in the advancement of
theoretical physics and chemistry. This methodological approach provides crucial insights into the quantum mechanical
behavior and interaction properties of fundamental particles.
Despite signicant progress in computational physics, challenges persist in the eective implementation of spectral
methods for such studies. Issues such as accurate wave function discretization, ecient Hamiltonian matrix management,
and precise eigenvalue calculation are critical for enhancing the delity of spectral analyses. These challenges have been
highlighted in recent studies which also demonstrate the potential of advanced computational methods in addressing these
complexities (Woywod et al. (2018), Iacob (2014)).
This research aims to bridge these gaps by developing a sophisticated computational framework for the spectral analysis
of the hydrogen atom. By employing Python and integrating cutting-edge techniques such as Chebyshev nodes for radial
discretization, ecient Hamiltonian matrix representation, and the Lanczos method for eigenvalue computation, this
study seeks to set a new standard in precision for quantum mechanical investigations. The successful implementation of
this approach could signicantly impact both theoretical understanding and practical applications in spectroscopy and
quantum technologies, underscoring the ongoing importance of spectral theory in contemporary scientic research.
Previous studies have addressed various aspects of spectral methods in quantum systems. For instance, the spectral
characterization of hydrogen-like atoms conned by oscillating systems has been investigated using ecient computational
methods Iacob (2014). Furthermore, the detailed analysis of the hydrogen atom’s spectrum has been fundamental to the
development of quantum mechanics laws Hnsch et al. (1979). Recently, pseudo-spectral methods for the description
of electronic wave functions in atoms and atomic ions have been developed Woywod et al. (2018). These backgrounds
highlight both advances and gaps in the implementation of spectral methods.
Despite these advances, there is a need for a methodology that more eectively integrates radial discretization, Hamiltonian
matrix management, and eigenvalue calculation. The lack of a comprehensive approach that simultaneously and eciently
addresses these aspects remains a challenge in the computational implementation of spectral methods in the hydrogen
atom.
This work aims to develop and implement an innovative computational approach for the spectral analysis of the hydrogen
atom using Python. It intends to combine radial discretization using Chebyshev nodes with ecient Hamiltonian matrix
representation and the Lanczos spectral method for precise eigenvalue calculation, thus addressing the current limitations
in the spectral study of the hydrogen atom.
In this sense, the objectives of the research are: Implement a radial grid using Chebyshev nodes for the ecient
discretization of the hydrogen atom; Develop an approach to eciently manage the non-zero values of the
Hamiltonian matrix, maximizing computational eciency; Apply the Lanczos spectral method for the precise and
ecient calculation of eigenvalues, facilitating a better understanding ofthe spectral properties of the hydrogen
atom.
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2. THEORETICAL FRAMEWORK
2.1 Spectral theory
2.1.1 Fundamentals of spectral theory
Spectral theory is an essential pillar in the mathematical study of linear systems, particularly in the context of operators in
Hilbert and Banach spaces. This theory has direct implications in quantum physics, especially in the analysis of quantum
systems such as the hydrogen atom. Spectral operators allow for a deep understanding of the properties and behavior of
these systems.
2.1.2 Applications in quantum physics
In quantum physics, spectral theory is extensively applied to analyze and understand the spectral of atoms and molecules.
The spectral structure of an atom, like the hydrogen atom, reveals fundamental information about its energy levels and
possible electronic transitions. This understanding is crucial for the development of quantum models and the interpretation
of physical phenomena at the atomic and molecular level.
2.2 Computational methods in quantum physics
2.1.1 Tools and programming languages
In the realm of quantum physics, computational advancements have provided essential tools for analyzing and modeling
complex quantum systems. Python, in particular, has emerged as a prominent language due to its exibility and the
availability of advanced scientic libraries such as NumPy and SciPy. These libraries facilitate the implementation of
quantum algorithms and the handling of complex numerical calculations.
Denition 1 (Hilbert space). A Hilbert space is a complete vector space equipped with an inner product that induces
a norm and a metric. In quantum physics, quantum states are modeled using vectors in a Hilbert space, where the inner
product represents the probability of transition between states.
Theorem 1 (Spectral theorem for compact operators). Let T be a compact and self-adjoint linear operator in a Hilbert
space . Then, T has a discrete set of eigenvalues {λ
n
}
∞
n =1
with corresponding orthonormal eigenvectors {e
n
}
∞
n=1
. Any
element can be expressed in terms of these eigenvectors as:
where denotes the coecient of f in the direction of e
n
.
Remarj 1. A detailed demonstration of this theorem can be found in classical texts on functional analysis. For further
reading and in-depth understanding, refer to works such as Reed and Simon’s “Methods of Modern Mathematical Physics
I: Functional Analysis” Reed & Simon (1980), Rudin’s “FunctionalAnalysis” Rudin (1991), and Conway’s “A Course in
Functional Analysis” Conway (1990).
2.2.2 Discretization and computational grids
Discretization in numerical analysis is the process of approximating a continuous function by a set of discrete values.
In quantum physics, this typically involves approximating wave functions or other physical quantities on a discrete
computational grid.
A computational grid is a discretely dened space where numerical solutions of physical equations are computed. In the
context of quantum systems, this grid could represent the spatial domain over which the quantum wave function is dened.
Theorem 2 (Spectral discretization). Let {ϕ
n
} be a set of orthogonal basis functions dened on a computational grid. Any
square-integrable function f(x), dened on this grid, can be approximated as:
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where c
n
are coecients determined by projecting f onto the basis functions ϕ
n.
The coecients c
n
are found by exploiting the orthogonality of the basis functions. For each basis function ϕ
n,
the coecient
c
n
is calculated using the inner product in the Hilbert space. Specically,
where the base is orthonormal in the interval (a, b) and the integrals are taken over the domain of f(x). This orthogonality
reduces computational complexity and ensures accuracy in the approximation of f(x). The summation
converges to the function f(x) in the limit as N → ∞, providing an ecient spectral representation of the function on the
computational grid.
A detailed exposition of this proof, along with comprehensive discussions on the underlying theoretical concepts, can be
found in the book Spectral Theory and Quantum Mechanics by Valter Moretti Moretti (2013). This reference provides a
profound insight into the spectral theory applied to quantum mechanics, which signicantly enhances the understanding
of the topics discussed herein.
2.3 Matrix of the hamiltonian
2.3.1 Representation and physical signicance
The Hamiltonian in quantum physics is a fundamental concept representing the total energy of a quantum system and
plays a central role in the Schrödinger equation.
Denition 2 (Hamiltonian operator). In quantum mechanics, the Hamiltonian operator H represents the total energy of
the system, combining kinetic T and potential V energies:
typically in the form:
where is the reduced Planck constant, m the particle’s mass, the Laplacian, and V (r) the potential energy.
2.3.2 Management of non-zero values in hamiltonian matrix
Ecient handling of non-zero values in large, sparse Hamiltonian matrices is crucial in quantum computations.
Theorem 3 (Sparse matrix eciency). Utilizing sparse matrix storage and algorithms enhances the computational
eciency of large, sparse Hamiltonian matrices, reducing memory and computational time.
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For more details related to this last theorem, see Woywod et al. (2018). Consider a sparse Hamiltonian matrix H of size
N × N with a sparsity pattern such that the number of non-zero elements is much less than N
2
. The goal is to demonstrate
that sparse matrix techniques enhance computational eciency.
Memory usage: In standard matrix storage, memory requirement is proportional to N
2
. For a sparse matrix, using formats
like Compressed Sparse Row (CSR), the memory requirement is proportional to the number of non-zero elements. Let
M be the number of non-zero elements, then memory usage in CSR format is approximately proportional to M, which is
signicantly less than N
2
for sparse matrices.
Computational time: Matrix operations such as multiplication or eigenvalue computation are more ecient in sparse
format. In the CSR format, matrix-vector multiplication complexity is approximately O(M), compared to O(N
2
) in dense
format. For eigenvalue computations, algorithms like Publicación Cuatrimestral. Vol. 9, No. 1, Enero/Abril, 2024,
Ecuador (p. 1 -17) 7 César Reinoso, Ramón Abancin the Lanczos method are optimized for sparse matrices, converging
in fewer iterations than would be required for a full, dense matrix.
Thus, using sparse matrix techniques, both memory usage and computational time are reduced, enhancing overall
computational eciency, which is crucial in handling large-scale quantum systems.
2.4 Lanczos spectral method
2.4.1 Description of the Lanczos method
The Lanczos method is a powerful algorithm in numerical linear algebra, eective for eigenvalues and eigenvectors of large
sparse matrices.
This iterative algorithm approximates the eigenvalues and eigenvectors of large sparse symmetric (or Hermitian) matrices,
constructing an orthogonal basis for the Krylov subspace and reducing the problem to a smaller tridiagonal matrix.
Theorem 4 (Convergence of the Lanczos method). The Lanczos method converges to accurate approximations of the largest
and smallest eigenvalues and their corresponding eigenvectors in fewer iterations than the matrix dimension.
The convergence involves the properties of the Krylov subspace and the orthogonality of Lanczos vectors. For detailed
demonstration, see “Numerical Linear Algebra” by Trefethen and Bau Trefethen & Bau (1997).
2.4.2 Applications and advantages in eigenvalue calculation
The Lanczos method is useful in quantum physics, particularly for eigenvalues of the hydrogen atom’s Hamiltonian.
Proposition 1 (Benets in quantum physics). The Lanczos method oers a computationally ecient approach to the eigenvalue
problem for the Hamiltonian in quantum physics, especially for large, sparse matrices.
This method’s ability to accurately approximate eigenvalues and eigenvectors with reduced computational burden is
advantageous for studying complex quantum systems.
Specically, the Lanczos method, a cornerstone in the computational implementation of spectral theory, excels in its
eciency due to its precise manipulation of Krylov subspaces to project large, sparse matrices into tridiagonal form. This
tridiagonalization is key to the method’s prowess, as it signicantly simplies the matrix structure, thereby streamlining
the calculation of eigenvalues and eigenvectors which are central to spectral analysis. The rened focus on transforming
the matrix to a tridiagonal matrix allows the use of optimized algorithms for eigenvalue computation, dramatically
reducing the computational burden. Such eciency is indispensable in spectral theory applications
where high-dimensional data and large matrix sizes are common, providing a robust, scalable solution that enhances
computational feasibility without compromising the accuracy essential for quantum mechanics and other physics-related
elds.
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2.5 Recent advances and current challenges
2.5.1 Advances in the study of the hydrogen atom
Recent advancements in the study of the hydrogen atom have been inuenced by developments in quantum chemical
approaches and molecular spectroscopy.
2.5.2 Challenges and areas for improvement
Ongoing challenges in applying spectral methods to quantum systems include managing the complexity in large
quantum systems. Further improvements in computational techniques are needed.
2.6 Mathematical and physical framework of the hydrogen atom
The Schrödinger equation is pivotal in quantum mechanics, providing profound insights into the behavior of
quantum systems, particularly the hydrogen atom. This equation is represented as follows:
(1)
where:
• denotes the reduced Planck constant.
• m represents the electron mass.
• symbolizes the Laplacian operator.
• ψ(r) is the electron’s wave function.
• V (r) signies the electric potential, particularly the Coulomb potential in a hydrogen atom.
• E is the total energy of the system.
The application of equation (1) to the hydrogen atom simplies due to its spherical symmetry, enabling the separation
of the wave function into radial and angular components. This aspect is thoroughly discussed in Griths’ work on
quantum mechanics Griths (2018).
2.6.1 Variable separation in the hydrogen atom
The unique symmetry of the hydrogen atom plays a critical role in the simplication of the Schrödinger equation. In
quantum mechanics, the hydrogen atom is modeled as a single electron orbiting a stationary nucleus, giving rise to
a spherically symmetric potential. This spherical symmetry allows for the separation of variables in the Schrödinger
equation, a method that simplies the complex threedimensional problem into more manageable one-dimensional
equations.
The separation of variables is achieved by expressing the electron’s wave function, ψ(r), as a product of two
functions: one depending solely on the radial coordinate, R(r), and the other on the angular coordinates, Y (θ, ϕ).
This is mathematically represented as:
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Here, R(r) describes how the wave function varies with distance from the nucleus, while Y (θ, ϕ), typically represented
by spherical harmonics, describes its variation with respect to the angular position. This separation signicantly
simplies the computational analysis of the hydrogen atom, allowing for the detailed study of its energy levels and
electron probability distributions, as elaborated in quantum physics literature Griths (2018).
2.6.2 Spectral numerical methods in quantum mechanics
The implementation of spectral numerical methods is crucial for solving the Schrödinger equation in a discrete
domain. This approach includes the discretization of space and the application of methods such as Fourier Transforms
or orthogonal polynomials to approximate wave functions and eigenvalues.
3. Numerical results and analysis
3.1 Radial mesh and Hamiltonian matrix analysis
The radial mesh, crucial for accurate quantum state representation, and the sparse nature of the Hamiltonian matrix are
illustrated in Figures 1 and 2 respectively.
Figure 1. Radial mesh using Chebyshev nodes.
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Figure 2. Non-zero values in the Hamiltonian matrix.
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3.2 Application of the Lanczos Method
The Lanczos method was applied to the Hamiltonian matrix to calculate its eigenvalues and eigenvectors. The wave
functions for the rst three quantum states are shown in the following gures.
Figure 3. Wave functions of dierent quantum states.
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The Lanczos method’s application to the Hamiltonian matrix reveals signicant insights into the quantum states of the
system. Figure 3 (Wave function of state 1) illustrates the wave function for the rst quantum state, characterized by a
high probability density near the origin, aligning with the theoretical expectations for ground state behavior in quantum
systems. This state represents the most stable conguration of the quantum system, having the lowest energy.
Moving to Figure 3 (Wave function of state 2), the wave function of the second state displays a single node, indicative of
the rst excited state. This nodal structure represents a zero probability region, a characteristic feature of quantum systems
where the particle is never found. The increased radial distribution compared to the rst state aligns with the higher energy
level of this state.
Lastly, Figure 3 (Wave function of state 3) shows the third state, which exhibits two nodes, corresponding to the pattern
of increasing nodes for higher energy levels. The spatial extent of the wave function, further from the origin, reects the
energy increase. These nodes provide deep insights into the spatial probability distribution of the quantum particle.
The Lanczos method eectively computes the eigenvalues and eigenvectors of the Hamiltonian ma trix, revealing critical
aspects of quantum state behaviors. The progression in the number of nodes and spatial extent of the wave functions across
these states oers a clear visualization of quantum mechanical principles and energy distribution in quantum systems.
3.3 Discussion
The Lanczos method’s implementation, as applied to the Hamiltonian matrix for quantum systems, has yielded valuable
insights into the spectral characteristics of atomic structures. The radial mesh visualization, shown in Figure 1, underscores
the ecacy of using Chebyshev nodes for discretizing quantum systems, a method corroborated by Iacob (2014). The
non-zero values of the Hamiltonian matrix, illustrated in Figure 2, reveal the sparse nature of quantum systems, aligning
with the studies in spectral theory Reed & Simon (1980); Rudin (1991); Taylor & Lay (2005).
The eigenvalues and eigenvectors obtained through the Lanczos method, depicted in Figure 3, oer a deeper understanding
of quantum states. The ground state, with its high probability density near the origin, reects foundational principles in
quantum mechanics Hnsch et al. (1979). The subsequent states, with their increasing nodes, align with the theoretical
predictions of quantum mechanics for higher energy states Woywod et al. (2018). This aligns with the spectral theory’s
assertion on the behavior of linear operators in Hilbert spaces Conway (1990); Dunford & Schwartz (1958).
The progression of the wave functions’ complexity with increasing quantum states serves as a vivid demonstration of
the underlying quantum mechanics principles, as well as the mathematical intricacies captured by the Lanczos method
Trefethen (1997); Trefethen & Bau (1997). These ndings not only validate the method’s computational eciency but
also its capability to reveal intricate details of quantum systems, oering a promising avenue for future research in
quantum physics and spectral graph theory Vadhan (2023).
4. Conclusions
This research has successfully met its primary goal of developing an innovative computational approach to the spectral
analysis of the hydrogen atom. Utilizing Python, the study combined radial discretization using Chebyshev nodes,
ecient management of the Hamiltonian matrix’s non-zero values, and the application of the Lanczos spectral method
for eigenvalue calculations.
The implementation of a radial grid using Chebyshev nodes, demonstrated in Figure 1, eectively discretized the hydrogen
atom. This approach provided a precise and ecient means to represent the quantum system, paving the way for more
accurate computational analysis. Furthermore, the strategy developed for handling the non-zero values of the Hamiltonian
matrix, as shown in Figure 2, optimized computational eciency. This was crucial in managing the sparse nature of the
matrix and aligns with current trends in spectral theory.
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Most importantly, the application of the Lanczos spectral method, illustrated through the eigenvalues and eigenvectors
in Figure ??, enabled a deeper understanding of the hydrogen atom’s spectral properties. This method provided precise
insights into the atom’s energy levels and probability distributions, showcasing the eectiveness of this computational
technique in quantum physics.
In conclusion, the project’s ndings signicantly contribute to our knowledge of quantum systems’ spectral characteristics.
The successful integration of these computational methods not only validated the study’s approach but also oers a
promising direction for future research in quantum physics and spectral graph theory. This study underscores the potential
of computational techniques in advancing our understanding of complex quantum phenomena.
5. Declaration of conict of interest of the authors
The authors declare no conict of interest.
6. Thanks
This work acknowledges the Mathematic degree program at Escuela Superior Politécnica de Chimborazo (ESPOCH) for
its unconditional support in carrying out this research.
7. References
Conway, J. B. (1990). A course in functional analysis. Springer-Verlag.
Dunford, N., & Schwartz, J. T. (1958). Linear operators, part ii: Spectral theory. Wiley Classics Library.
Griths, D. J. (2018). Introduction to quantum mechanics (3rd ed.). Cambridge University Press.
Hnsch, T. W., Schawlow, A. L., & Series, G. W. (1979). The spectrum of atomic hydrogen. Scientic American, 240(3),
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Revista de la Facultad de Ciencias Básicas
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DOI: 10.33936/revbasdelaciencia.v9i1.6403
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Author Contributions
Autor Contribución
César Daniel Reinoso Reinoso Bibliographic search, design, writing and revision of the article.
Ramón Antonio Abancin Ospina Conception, Methodology, Analysis, review and writing of the
article.
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MÉTODOS ESPECTRALES SOBRE EL ÁTOMO DE HIDRÓGENO
César Reinoso, Ramón Abancin
revista.bdlaciencia@utm.edu.ec
Vol. 9, Núm. 1 (15-28): Enero-Abril, 2024
Revista de la Facultad de Ciencias Básicas
ISSN 2588-0764
Bases de la Ciencia
DOI: 10.33936/revbasdelaciencia.v9i1.6403
A. APPENDIX
A.1 Python mode for Lanczos method and analysis
Listing 1: Python code for implementing the Lanczos method and plotting wave functions.
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https://revistas.utm.edu.ec/index.php/Basedelaciencia
Portoviejo - Manabí - Ecuador
BASES DE LA CIENCIA
Revista Científica
Facultad de Ciencias Básicas
revista.bdlaciencia@utm.edu.ec
Vol. 9, Núm. 1 (15-28): Enero-Abril, 2024
Revista de la Facultad de Ciencias Básicas
Bases de la Ciencia
DOI: 10.33936/revbasdelaciencia.v9i1.6403
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