In physics, many problems arise in the form of boundary value problems in second order ordinary differential equations. We are discussing here the Matrix Variational Method, as an efficient approach to bound state eigenproblems [1--8], proposed by the author starting in 1977 and used in top peer-reviewed literature in physics, in general [1,6,7] and in the calculation of scaling laws of Rydberg atoms [2], bound states of QM systems [3], bound states of three quarks [4,8], and other areas, such as [5].
We will also use this RG space in a new way, to conduct an open course, as a discussion. This course is physically offered at the same time, in Pasadena, CA.
The objective here is to present the topic as a method in mathematics, for second-year students in college, generally when they see differential equations, not just the epsilons and deltas of calculus, but the more advanced tools and intuition used in physics and maths.
This discussion will aim, as much as possible, to be free of the original connection to physics, in order to be more easily used in other disciplines. It represents the “translation” of a method in physics to mathematics, for general use, while benefiting from the physical intuition that started it.
We will use the theorem that says, “Any second order linear operator can be put into the form of the Sturm-Liouville operator,” and treat the Sturm-Liouville operator in closed-form. This will be done not by using eigenfunctions of any expansion, but an expansion that already obeys the boundary conditions for each case and provides a closed-form expression, which we will calculate following [1-8].
Contributions, and other examples, are welcome.
REFERENCES
[1] Ed Gerck, A. B. d'Oliveira, Matrix-Variational Method: An Efficient Approach to Bound State Eigenproblems, Report number: EAV-12/78, Laboratorio de Estudos Avancados, IAE, CTA, S. J. Campos, SP, Brazil. Copy online at https://www.researchgate.net/publication/286625459_Matrix-Variational_Method_An_Efficient_Approach_to_Bound_State_Eigenproblems
[2] Jason A C Gallas, Ed Gerck, Robert F O'Connell, Scaling Laws for Rydberg Atoms in Magnetic Fields, Physical Review Letters 50(5):324-327, Jan 1983. Copy online at
https://www.researchgate.net/publication/243470610_Scaling_Laws_for_Rydberg_Atoms_in_Magnetic_Fields
[3] Ed Gerck, Jason A C Gallas, Augusto. B. d'Oliveira, Solution of the Schrödinger equation for bound states in closed form, Physical Review A 26:1(1), June 1982. Copy online at
https://www.researchgate.net/publication/236420748_Solution_of_the_Schrodinger_equation_for_bound_states_in_closed_form
[4] A. B. d'Oliveira, H. F. de Carvalho, Ed Gerck, Heavy baryons as bound states of three quarks, Lettere al Nuovo Cimento 38(1):27-32, Sep 1983. Copy online at
https://www.researchgate.net/publication/243613137_Heavy_baryons_as_bound_states_of_three_quarks
[5] Ed Gerck, A. B. d'Oliveira, The non-relativistic three-body problem with potential of the form K1r^n + K2/r + C, Report number: EAV-11/78, Laboratorio de Estudos Avancados, IAE, CTA, S. J. Campos, SP, Brazil, Nov1978. Copy online at
https://www.researchgate.net/publication/286640675_The_non-relativistic_three-body_problem_with_potential_of_the_form_K1rn_K2r_C
[6] Ed Gerck, Augusto Brandão d'Oliveira, Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schroedinger equation, Journal of Computational and Applied Mathematics 6(1):81-82, Mar 1980. Copy online at
Article Continued fraction calculation of the eigenvalues of tridiag...
[7] Ed Gerck, A. B. d'Oliveira, Jason A C Gallas, New Approach to Calculate Bound State Eigenvalues, Revista Brasileira de Ensino de Física, 13(1):183-300, Jan 83. Copy online at
Article New Approach to Calculate Bound State Eigenvalues
[8] Ed Gerck, A. B. d'Oliveira, The logarithmic and the square-root potential as confining potentials for quarks, Report number: EAV Report 02/79, Laboratorio de Estudos Avancados, IAE, CTA, S. J. Campos, SP, Brazil. Copy online at
Technical Report The logarithmic and the square-root potential as confining p...
The discussion will try a suggested schedule, by one or more student presentations.
1. By October 20 2018: A preamble, with the Introduction, Intuitive Model, Mathematical Model with Alternatives for Solution, and References. References include Chapter 6: "Sturm-Liouville Eigenvalue Problems" at
http://people.uncw.edu/hermanr/mat463/ODEBook/Book/SL.pdf
and others, supporting various solution methods.
2. By November 8, 2018: Finalize the Mathematical Model using the matrix-variational solution [1-5] op. cit., in support of the Intuitive Model, and with suggestions for teaching physics, including quantum mechanics, motivating that there must be a solution valid for the given null boundary condition at Infinity (common in physics).
3. By November 30, 2018: present work in LaTeX, PDF, in paper format, including research topics that, hopefully, will develop in the work. The work intends to, at least, revisit and extend the matrix-variational method proposed [1-5], in physics.
4. By December 20, 2018: Final form presentations by one or more students, PDF in LaTeX.
I will be providing comments in-between.
Cheers, Ed Gerck
I have completed my first report on the program above. It is available for comments at
Preprint On the Matrix Variational Method (MVM), 1
Dear Ed,
As you better know old wine is better, so old works bring us always positive sentiments.
So, after so many years, tell me something:
- is the use of finite difference methods absolutely necessary after having today's available symbolics computing tools?
Demetris: Yes! The MVM, for example, produces simple closed-form expressions, not an infinity of series expressions, or numbers. The WKB method is used in QM, for the same reason. Variational methods, in general, or such as the principle of virtual work.
The point in the MVM is approximating the operator, not the solution. The solution is exact. The operator is a finite difference, not a differential, this works for the set used, with a trivial to diagnolize matrix, bidiagonal.
This is critical for scaling laws, and other uses, such as producing intuition, where the behavior comes from a very compact result, as cited in the text and the references in https://www.researchgate.net/publication/328416858_On_the_Matrix_Variational_Method_MVM_1
Demetris,
Thank you for your comment. It reminds me of a story by Isaac Asimov, where the word 'computer', originally meant for humans, became used only by machines, and so no one would even know how to add two numbers themselves, without a computer, until... they rediscovered arithmetic.
So, it seems important, and I will deal with it in the second report, but it includes a misconception. Mathematics is not computing, and more can be learned by understanding how things work, such as scaling laws, instead of blindly solving a differential equation. I see that several times.
So essentially it is similar to solving the problem of ODE:
L(D)y=f(x)
and after factoring the L(D) in
L(D)=(D-p1)(D-p2)...(D-pn)
then we proceed to solve easy problems?
I have completed my second report on the program above. It is available for comments at
Preprint On a Variational Method for Solving the Sturm-Liouville Diff...
I have completed my third report on the program above. It answers five representative questions, including from my online collaboration with this group.
The report explains more in depth both the MVM and other methods to solve the Sturm-Liouville eigenproblem.
The report is available for comments at
Preprint On a Variational Method for Solving the Sturm-Liouville Diff...
I have completed my final report on the program above. It is available for comments at
Preprint On the Matrix-Variational Method (MVM) for Solving the Sturm...
It turns out that matrix multiplication becomes similar to scalar multiplication for the eigenvectors, reproducing the eigenvalues (i.e., this is how we can see CH). Someone might think it is strange to use eigenvalues and eigenvectors to solve a matrix equation, but we find this metaphor (with scalar multiplication) to be illuminating -- it is not just a mathematical procedure.
The MVM can be extended to finding potentially new quantum models in day-to-day occurrences such as numerical sequences and measurement of time series.
Many systems obey known numerical sequences, the question here is if there is a quantum model that fits the observation, and secondly, what, if any, insight can we get from such a physics model?
Please take a look at this extension at https://www.researchgate.net/post/Using_inverse_eigenvalue_solutions_to_represent_numerical_sequences_and_time_series
Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behaviour of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps.
in https://arxiv.org/abs/0804.2605
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