Harmonic analysis | Theta functions | Operator theory | Representation theory
In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself. The contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended. (Wikipedia).
Physics - Ch 66 Ch 4 Quantum Mechanics: Schrodinger Eqn (45 of 92) Quantum Nature of Oscillator 1
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain the quantum mature of the oscillator. I will explain the step-function that represent the energy differences between different energy states. The change of the energy can only happen 1 step an
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From playlist Physics Demonstrations
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From playlist How to use an Oscilloscope
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From playlist THE "WHAT IS" PLAYLIST
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The main point of this demonstration is to hear the beats. It may be desirable, however, for the students to also have a visual display of what is happening to cause the beats. This is the purpose of the oscilloscope.
From playlist Beats
Demonstrating the phenomenon of beats on the oscilloscope (normal speed)
The main point of this demonstration is to hear the beats. It may be desirable, however, for the students to also have a visual display of what is happening to cause the beats. This is the purpose of the oscilloscope.
From playlist Beats
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From playlist MECHANICS
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