Rigged Hilbert Space X: Eq (1) and (2)

TTT · 2019-11-28T06:27:44+00:00

X=e+or-kx (1) =∫-∞∞X*(x)Φ(x)dx (2) where Φ(x) satisfies the following. ∫-∞∞|Φ(x)|2(1+|x|)ndx is finte if n=0, 1, 2,...

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Start date Nov 28, 2019
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Hilbert Hilbert space Space

In summary, a Rigged Hilbert Space is an extension of the traditional Hilbert Space used in quantum mechanics, allowing for a more flexible representation of physical states. It utilizes equations such as Eq (1) and (2) to calculate properties and interactions within the space, making it useful for studying quantum mechanics. It differs from other types of Hilbert Spaces by including a wider range of functions and states, making it more versatile. Its applications include studying scattering processes, resonances, and quantum field theory. However, it has limitations and may not be applicable to all problems in quantum mechanics and requires a certain level of mathematical understanding.

Nov 28, 2019

TTT

TL;DR Summary: Does the exponential function (1), where k >0, belong to the Rigged Hilbert Space and Why? The Rigged Hilbert Space is defined as the set of functions X(x) such that Inner product (2) is finite. * on (2) refers to a conjugate

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X=e^+or-kx (1)
<X(x)|Φ(x)>=∫_-∞^∞X*(x)Φ(x)dx (2)
where Φ(x) satisfies the following.
∫_-∞^∞|Φ(x)|²(1+|x|)ⁿdx is finte if n=0, 1, 2,...

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Nov 28, 2019

dextercioby

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1. This looks much better, if you learned how to use LaTex. There are tutorials here. Just look for "LaTex typesetting"
2. It looks suspiciously like a course homework question. So according to the PF Hw policy, what are your ideas on the problem?

1. What is a Rigged Hilbert Space?

A Rigged Hilbert Space is a mathematical concept used in quantum mechanics to describe the state of a quantum system. It is a generalization of a standard Hilbert space, which is a mathematical structure used to represent the state of a classical system. A Rigged Hilbert Space includes both the physical states of a system as well as the unphysical states, known as generalized states, which are used to describe the behavior of the system.

2. What is Eq (1) and Eq (2) in Rigged Hilbert Space X?

Eq (1) and Eq (2) refer to two equations that are used to describe the behavior of a quantum system in a Rigged Hilbert Space X. These equations are known as the Schrödinger equation and the Heisenberg equation, respectively. The Schrödinger equation describes how the state of a system evolves over time, while the Heisenberg equation describes the evolution of observables, such as position and momentum, over time.

3. How is Rigged Hilbert Space X used in quantum mechanics?

Rigged Hilbert Space X is used in quantum mechanics as a mathematical framework to describe the state and behavior of quantum systems. It allows for a more complete and accurate description of these systems by including both physical and unphysical states. This framework is particularly useful in describing systems with infinite degrees of freedom, such as particles in a continuous space.

4. What are the applications of Rigged Hilbert Space X?

Rigged Hilbert Space X has many applications in quantum mechanics, including the study of quantum field theory, quantum optics, and quantum information theory. It is also used in the development of new quantum technologies, such as quantum computers and quantum cryptography. Additionally, Rigged Hilbert Space X has applications in other fields, such as signal processing and image reconstruction.

5. Are there any limitations to using Rigged Hilbert Space X?

Like any mathematical framework, Rigged Hilbert Space X has its limitations. It can be challenging to apply to systems with complex interactions or non-linear behavior. Additionally, the use of unphysical states in this framework can be controversial and may not always accurately describe the behavior of a system. However, Rigged Hilbert Space X remains a valuable tool in understanding and predicting the behavior of quantum systems.