scholarly journals Geometric Cauchy-Schwarz inequality

2020 ◽  
Author(s):  
wang
Keyword(s):  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Schwarz Inequality ◽  
Quantum Method ◽  
Environment Variable

In this paper, using the generalized geometric commutator and geomutatorto develop the geometric anticommutator and anti-geomutator, then we strictly prove geometric Cauchy-Schwarz inequality in quantum method as a generalization of the Cauchy-Schwarz inequality based on the Hermitian operator. It certainly demonstrates a fact that environment variable is unavoidable for the revision of the Cauchy-Schwarz inequality for the quantum mechanics.

scholarly journals Time-Dependent Pseudo-Hermitian Hamiltonians and a Hidden Geometric Aspect of Quantum Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (4) ◽  
pp. 471 ◽  
Author(s):  
Ali Mostafazadeh
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Quantum Systems ◽  
Time Dependent ◽  
Inner Product ◽  
Geometric Framework ◽  
Heisenberg Picture ◽  
Metric Operator ◽  
Recent Developments

A non-Hermitian operator H defined in a Hilbert space with inner product ⟨ · | · ⟩ may serve as the Hamiltonian for a unitary quantum system if it is η -pseudo-Hermitian for a metric operator (positive-definite automorphism) η . The latter defines the inner product ⟨ · | η · ⟩ of the physical Hilbert space H η of the system. For situations where some of the eigenstates of H depend on time, η becomes time-dependent. Therefore, the system has a non-stationary Hilbert space. Such quantum systems, which are also encountered in the study of quantum mechanics in cosmological backgrounds, suffer from a conflict between the unitarity of time evolution and the unobservability of the Hamiltonian. Their proper treatment requires a geometric framework which clarifies the notion of the energy observable and leads to a geometric extension of quantum mechanics (GEQM). We provide a general introduction to the subject, review some of the recent developments, offer a straightforward description of the Heisenberg-picture formulation of the dynamics for quantum systems having a time-dependent Hilbert space, and outline the Heisenberg-picture formulation of dynamics in GEQM.

scholarly journals Equivalent Hermitian operator from supersymmetric quantum mechanics

2010 ◽  
Vol 374 (19-20) ◽  
pp. 1962-1965 ◽  
Author(s):  
Boris F. Samsonov ◽  
V.V. Shamshutdinova ◽  
A.V. Osipov
Keyword(s):  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Supersymmetric Quantum Mechanics

scholarly journals Modes and Noise Propagation in Phase Space

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
J. S. Ben-Benjamin ◽  
L. Cohen
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Dispersion Relation ◽  
Wigner Distribution ◽  
Hermitian Operator ◽  
Type Equation ◽  
Hamiltonian Operator ◽  
Dispersive Media ◽  
Phase Space Methods ◽  
Complex Dispersion

AbstractWe show that phase space methods developed for quantum mechanics, such as the Wigner distribution, can be effectively used to study the evolution of nonstationary noise in dispersive media. We formulate the issue in terms of modes and show how modes evolve and how they are effected by sources.We show that each mode satisfies a Schrödinger type equation where the “Hamiltonian” may not be Hermitian. The Hamiltonian operator corresponds to dispersion relationwhere thewavenumber is replaced by the wavenumber operator. A complex dispersion relation corresponds to a non Hermitian operator and indicates that we have attenuation. A number of examples are given.

scholarly journals Time Observables in a Timeless Universe

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 354
Author(s):  
Tommaso Favalli ◽  
Augusto Smerzi
Keyword(s):  
Quantum Mechanics ◽  
Energy Spectrum ◽  
Hermitian Operator ◽  
Dynamical Evolution ◽  
Emergent Property ◽  
Time Operator ◽  
Static Universe ◽  
Unequally Spaced ◽  
Energy Ratios ◽  
Time In Quantum Mechanics

Time in quantum mechanics is peculiar: it is an observable that cannot be associated to an Hermitian operator. As a consequence it is impossible to explain dynamics in an isolated system without invoking an external classical clock, a fact that becomes particularly problematic in the context of quantum gravity. An unconventional solution was pioneered by Page and Wootters (PaW) in 1983. PaW showed that dynamics can be an emergent property of the entanglement between two subsystems of a static Universe. In this work we first investigate the possibility to introduce in this framework a Hermitian time operator complement of a clock Hamiltonian having an equally-spaced energy spectrum. An Hermitian operator complement of such Hamiltonian was introduced by Pegg in 1998, who named it "Age". We show here that Age, when introduced in the PaW context, can be interpreted as a proper Hermitian time operator conjugate to a "good" clock Hamiltonian. We therefore show that, still following Pegg's formalism, it is possible to introduce in the PaW framework bounded clock Hamiltonians with an unequally-spaced energy spectrum with rational energy ratios. In this case time is described by a POVM and we demonstrate that Pegg's POVM states provide a consistent dynamical evolution of the system even if they are not orthogonal, and therefore partially undistinguishables.

scholarly journals Quantity in Quantum Mechanics and the Quantity of Quantum Information

2021 ◽  
Author(s):  
Vasil Dinev Penchev
Keyword(s):  
Quantum Mechanics ◽  
Quantum Information ◽  
Density Distribution ◽  
Probability Density ◽  
Hermitian Operator ◽  
Space Time ◽  
Complex Hilbert Space ◽  
Probability Density Distribution ◽  
Classical Quantum ◽  
Definition Of

The paper interprets the concept “operator in the separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined in the “classical” quantum mechanics) by that of “quantum information”. As far as wave function is the characteristic function of the probability (density) distribution for all possible values of a certain quantity to be measured, the definition of quantity in quantum mechanics means any unitary change of the probability (density) distribution. It can be represented as a particular case of “unitary” qubits. The converse interpretation of any qubits as referring to a certain physical quantity implies its generalization to non-Hermitian operators, thus neither unitary, nor conserving energy. Their physical sense, speaking loosely, consists in exchanging temporal moments therefore being implemented out of the space-time “screen”. “Dark matter” and “dark energy” can be explained by the same generalization of “quantity” to non-Hermitian operators only secondarily projected on the pseudo-Riemannian space-time “screen” of general relativity according to Einstein's “Mach’s principle” and his field equation.

scholarly journals On Some Consequences of the Functional Generalization of the Parallelogram Identity

2014 ◽  
Vol 47 (2) ◽  
Author(s):  
Maciej J. Maczyński
Keyword(s):  
Quantum Mechanics ◽  
Functional Form ◽  
Schwarz Inequality ◽  
Vector Spaces ◽  
Complex Vector ◽  
Standard Structure ◽  
Heisenberg's Uncertainty Principle ◽  
Normed Vector ◽  
Mathematics And Physics

AbstractThe aim of this paper is to unify the partial results, which up to now, have been dispersed in various publications in order to show the importance of the functional form of parallelogram identity in mathematics and physics. We study vector spaces admitting a real non-negative functional which satisfies an identity analogous to the parallelogram identity in normed vector spaces. We show that this generalized parallelogram identity also implies an equality analogous to the Cauchy-Schwarz inequality. We study the consequences of this identity in real and complex vector spaces, in generalized Riesz spaces and in abelian groups. We give a physical interpretation to these results. For vector spaces of observables and states, we show that the parallelogram identity implies an inequality analogous to Heisenberg’s uncertainty principle (HUP), and we show that we can obtain the standard structure of quantum mechanics from the parallelogram identity, without assuming from the beginning the HUP. The role of complex numbers in quantum mechanics is discussed.

scholarly journals Can The Spectra Of Hermitian Operator Be Invariant Under x ⇀ ↽ p ? :Case Study :1D General

2018 ◽  
Vol 14 (1) ◽  
pp. 5326-5330
Author(s):  
Biswanath Rath
Keyword(s):  
Quantum Mechanics ◽  
Hermitian Operator ◽  
Spectral Invariance

We address an intriguing question on spectral invariance in quantum mechanics onexchange of co-ordinate and momentum x ⇀ ↽ p considering general oscillator as anexample.Key words:exchange of co-ordinate and momentum , spectral invariance , generaloscillator .

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