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Rigorous Bounds on Transmission, Reflection, and Bogoliubov Coefficients

Petarpa Boonserm

Published Jan 1, 2009• 14 citations
Publisher

Abstract

This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: Bounding the Bogoliubov coefficients; Bounding the greybody factors for Schwarzschild black holes; Transformation probabilities and the Miller-Good transformation; Analytic bounds on transmission probabilities.

DOI
10.26686/wgtn.16959334

Related papers

Rigorous Bounds on Transmission, Reflection, and Bogoliubov Coefficients

Petarpa Boonserm

This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: Bounding the Bogoliubov coefficients; Bounding the greybody factors for Schwarzschild black holes; Transformation probabilities and the Miller-Good transformation; Analytic bounds on transmission probabilities.

•Jan 1, 2009•22 citations

Rigorous bounds on Transmission, Reflection, and Bogoliubov coefficients

Petarpa Boonserm

This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: 1) Bounding the Bogoliubov coefficients; 2) Bounding the greybody factors for Schwarzschild black holes; 3) Transformation probabilities and the Miller--Good transformation; 4) Analytic bounds on transmission probabilities.

arXiv:0907.0045•Jul 1, 2009•0 citations

Transmission probabilities and the Miller–Good transformation

Petarpa Boonserm, Matt Visser

Transmission through a potential barrier, and the related issue of particle production from a parametric resonance, are topics of considerable general interest in quantum physics. The authors have developed a rather general bound on quantum transmission probabilities, and recently applied it to bounding the greybody factors of a Schwarzschild black hole. In the current paper, we take a different tack—we use the Miller–Good transformation (which maps an initial Schrödinger equation to a final Schrödinger equation for a different potential) to significantly generalize the previous bound.

arXiv:0808.2516•Journal of Physics A Mathematical and Theoretical•Dec 11, 2008•33 citations

Bounding the Bogoliubov coefficients

Petarpa Boonserm, Matt Visser

While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [Phys. Rev. A 59 (1999) 427–438], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2 × 2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.

•Jan 1, 2008•52 citations

One Dimensional Scattering Problems: A Pedagogical Presentation of the Relationship between Reflection and Transmission Amplitudes

Petarpa Boonserm, Matt Visser

In this article we provide a pedagogical introduction to scattering theory in one space dimension. This is an elegant topic that is mathematically simple and physically transparent. We shall apply the Schrodinger equation to a generic system to identify the Bogoliubov coefficients. Furthermore, we shall then derive a number of significant relationships between reflection and transmission amplitudes.

Thai Journal of Mathematics•Apr 15, 2012•11 citations

One Dimensional Scattering Problems: A Pedagogical Presentation of the Relationship between Reflection and Transmission Amplitudes: Petarpa Boonserm, Matt Visser

Support Team

In this article we provide a pedagogical introduction to scattering theory in one space dimension. This is an elegant topic that is mathematically simple and physically transparent. We shall apply the Schrodinger equation to a generic system to identify the Bogoliubov coefficients. Furthermore, we shall then derive a number of significant relationships between reflection and transmission amplitudes.

Thai Journal of Mathematics•Dec 31, 2010•0 citations
This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: Bounding the Bogoliubov coefficients; Bounding the greybody factors for Schwarzschild black holes; Transformation probabilities and the Miller-Good transformation; Analytic bounds on transmission probabilities.
This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: Bounding the Bogoliubov coefficients; Bounding the greybody factors for Schwarzschild black holes; Transformation probabilities and the Miller-Good transformation; Analytic bounds on transmission probabilities.
This thesis describes the development of some basic mathematical tools of wide relevance to mathematical physics. Transmission and reflection coefficients are associated with quantum tunneling phenomena, while Bogoliubov coefficients are associated with the mathematically related problem of excitations of a parametric oscillator. While many approximation techniques for these quantities are known, very little is known about rigorous upper and lower bounds. In this thesis four separate problems relating to rigorous bounds on transmission, reflection and Bogoliubov coefficients are considered, divided into four separate themes: 1) Bounding the Bogoliubov coefficients; 2) Bounding the greybody factors for Schwarzschild black holes; 3) Transformation probabilities and the Miller--Good transformation; 4) Analytic bounds on transmission probabilities.
Transmission through a potential barrier, and the related issue of particle production from a parametric resonance, are topics of considerable general interest in quantum physics. The authors have developed a rather general bound on quantum transmission probabilities, and recently applied it to bounding the greybody factors of a Schwarzschild black hole. In the current paper, we take a different tack—we use the Miller–Good transformation (which maps an initial Schrödinger equation to a final Schrödinger equation for a different potential) to significantly generalize the previous bound.
While over the last century or more considerable effort has been put into the problem of finding approximate solutions for wave equations in general, and quantum mechanical problems in particular, it appears that as yet relatively little work seems to have been put into the complementary problem of establishing rigourous bounds on the exact solutions. We have in mind either bounds on parametric amplification and the related quantum phenomenon of particle production (as encoded in the Bogoliubov coefficients), or bounds on transmission and reflection coefficients. Modifying and streamlining an approach developed by one of the present authors [Phys. Rev. A 59 (1999) 427–438], we investigate this question by developing a formal but exact solution for the appropriate second-order linear ODE in terms of a time-ordered exponential of 2 × 2 matrices, then relating the Bogoliubov coefficients to certain invariants of this matrix. By bounding the matrix in an appropriate manner, we can thereby bound the Bogoliubov coefficients.