Abstract
This is the first one of two companion papers focusing on the establishment of a new path for the expectation value dynamics of the quantum mechanical operators. The main goal of these studies is to do quantum mechanics without explicitly solving Schrödinger wave equation, in other words, without using wave functions except their initially given forms. This goal is achieved by using Ehrenfest theorem and utilizing probabilistic evolution approach (PEA). PEA, first introduced by Metin Demiralp, is a method providing solutions to the nonlinear ordinary differential equations by transforming them to a set of linear ODEs at the cost of denumerably infinite dimensionality. It is recently shown that this method produces analytic solutions, if the initial conditions are given appropriately at some special cases. However, generalization of these conditions to the quantum mechanical applications is not straightforward due to the dispersion of the quantum mechanical systems. For this purpose, multivariate moment problems for the integral representation of the Kronecker power series are introduced and then solved yielding to more specific and precise convergence analysis for the quantum mechanical applications.
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Notes
At this point it is better to emphasize on how the mathematical object “Sequence” is defined. It is known as “A list of elements”. It is not a set because the repetition of the elements is allowed and, beyond that, the elements should be ordered. The general tendency is to use same type elements, that is, the objects sharing at least one property like in positive integers, rational numbers, and so on. Here, the shared property is the global definition, “the integral of a given so-called state vector’s Kronecker powers under a generating function factor”.
The true definition of Hankel matrices requires the equivalence of all elements in its each anti-diagonal. It is satisfied if the matrix \(\mathbf {x}\) becomes a one-element vector, mainly scalar. The multivariance destroys the equivalence amongst the antidiagonal elements. However, we still keep the word “Hankel” in this matrix to recall the inspirations from the Hankel matrices even though we somehow emphasize on the discrimination by using the word “Extended” in the name of present case matrices.
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Acknowledgments
The first author thanks to The Scientific and Technological Research Council of Turkey (TUBITAK) for their support during his Ph.D. studies.
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Ayvaz, M., Demiralp, M. Probabilistic evolution approach to the expectation value dynamics of quantum mechanical operators, part I: integral representation of Kronecker power series and multivariate Hausdorff moment problems. J Math Chem 52, 2161–2182 (2014). https://doi.org/10.1007/s10910-014-0371-8
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DOI: https://doi.org/10.1007/s10910-014-0371-8
Keywords
- Probabilistic evolution approach
- Quantum mechanics
- Expectation value dynamics
- Hausdorff moment problem
- Kronecker power series
- Ordinary differential equations