BSAC Seminar: Arcsine Law for Quantum Harmonic Oscillators

March 20, 2012

Prof. Hayato Saigo

Nagahama Institute of Bio-Science and Technology, Japan
March 20, 2012 | 12:00 to 01:00 | 540 Cory Hall, DOP Center Conference Room
Host: Al Pisano

There is one mysterious probability law: The Arcsine Law. It has very different features in comparison to famous normal distribution (Gaussian). For instance, it is bounded and “polarized” - the probability around the bounds is large while probability density around expectation value is small! Nevertheless, the Arcsine Law is not just a strange toy for mathematicians. It is, and will be, so important for science and technology. It appears in relation with Brownian motions or random walks, quantum walks, etc. Moreover, we have recently shown that the Arcsine Law will appear as the law for “high energy quantum harmonic oscillators” by using quite simple framework of “Noncommutative Probability,” which is a unification of probability theory and quantum theory. Quantum version of harmonic oscillators (e.g., oscillating atoms in crystal) can be described by using two operators which increase/decrease the energy level. As the energy level increases, how do the quantum oscillators behave? We can prove that the behavior will be described by the Arcsine Law by using a version of “Binomial expansion.” In this seminar, we will see the mechanism is nothing but a rigorous realization for “Quantum-Classical Correspondence.”

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Jonathan Candelaria
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