Author Archives: Eric A. Galapon

I think this one is beautiful

By | September 14, 2015
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Beauty is one of those abstractions that is difficult to define or quantify. But we know it when we see one. I arrived at the following integral trying to make sense of some divergent integrals: \begin{equation} \frac{1}{2}\int_{-\infty}^{\infty} \left(\cos\left(\frac{t^{-1}-t}{2}\right) – t \sin\left(\frac{t^{-1}-t}{2}\right)\right) (1+t^2)^{-1} \, \mbox{d}t = \frac{\pi}{\mbox{e}} .\nonumber \end{equation} There is manifest symmetry in the integrand… Read More: I think this one is beautiful »

Who is afraid of divergent integrals?

By | July 29, 2015
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We, physicists, are obviously not afraid. Our literature, from our notes to our papers to our books, is replete with them. We are not at all embarrassed by our seemingly wanton disregard to mathematical rigor in our mathematical acrobatics that often lead to our ill-defined or infinite-valued divergent integrals. Our enduring affair with such almost… Read More: Who is afraid of divergent integrals? »

Quantum Systems and their Hilbert Spaces

By | July 12, 2015
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What is a Quantum System A quantum system is traditionally defined as an object or a collection of objects in the scale of the electron or of the atoms, the description of which is governed by the laws of quantum mechanics. However, it has become increasingly evident that the grips of quantum mechanics extend much… Read More: Quantum Systems and their Hilbert Spaces »

Super-Hilbert spaces and super-operators

By | June 17, 2015
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Introduction The dynamics of a quantum system plays out in a Hilbert space $\mathcal{H}$, which we, for simplicity’s sake, take here to be finite dimensional. In general the state of a closed quantum system, i.e. a system that does not exchange energy and information with another quantum system, is given by a vector $|\psi\rangle$ in… Read More: Super-Hilbert spaces and super-operators »

Explainer Part 1: “Exactly decohering quantum measurement without environment” E.A. Galapon, arXiv:1505.06908, 26 May 2015.

By | June 2, 2015
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The paper deals with the fundamental question of whether an environment is necessary or not to enforce quantum decoherence on quantum measuring instruments. The paper gives an unambiguous answer to the question as summarized by its Abstract: Current quantum orthodoxy claims that the statistical collapse of the wave-function arises from the interaction of the measuring… Read More: Explainer Part 1: “Exactly decohering quantum measurement without environment” E.A.… »

An objection of Asher Peres to quantizing classical time observables

By | May 9, 2015
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One of the modern books in quantum mechanics that I highly recommend to anyone is the book by the late Asher Peres—Quantum Theory: Concepts and Methods. Peres was one of the leading researchers in the foundations of quantum mechanics and a pioneer in quantum information theory. His keen understanding of quantum mechanics is clearly manifested… Read More: An objection of Asher Peres to quantizing classical time observables »

Are there countable number of quantum systems?

By | May 9, 2015
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Quantum mechanics demands that every quantum system corresponds to a separable Hilbert space. The separability condition requires that the Hilbert space possesses a countable set of basis vectors. A collection of objects or a set is called countable if it has a finite number of members or elements; or, in the case of infinite number… Read More: Are there countable number of quantum systems? »

Explainer: “The Bender-Dunne basis operators as Hilbert space operators”, Bunao and Galapon, J. Math. Phys. 55, 022102 (2014)

By | April 29, 2015
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Introduction We, physicists, work with our operators very differently from mathematicians. We are unencumbered by rigor and our mathematics proceeds with the motivation to forge ahead to gain insight as quickly as possible. Only later when our results are meaningful that we bother to step back and ask ourselves if our steps in reaching our… Read More: Explainer: “The Bender-Dunne basis operators as Hilbert space operators”, Bunao… »

The Hilbert space of quantum mechanics

By | April 29, 2015
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Introduction A system is an object or a collection of objects whose properties are subject to observation and measurement. The state of a given system is the minimum collection of information that completely describes the system. For a classical particle, its state is specified by giving its position and momentum at any given time. The… Read More: The Hilbert space of quantum mechanics »