scholarly journals Quantum Harmonic Analysis of the Density Matrix

Quanta ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 74 ◽  
Author(s):  
Maurice A. De Gosson
Keyword(s):  
Density Matrix ◽  
Harmonic Analysis ◽  
Quantum Mechanical ◽  
Mixed States ◽  
Density Matrices ◽  
Planck’S Constant ◽  
Planck's Constant

We will study rigorously the notion of mixed states and their density matrices. We will also discuss the quantum-mechanical consequences of possible variations of Planck's constant h. This review has been written having in mind two readerships: mathematical physicists and quantum physicists. The mathematical rigor is maximal, but the language and notation we use throughout should be familiar to physicists.Quanta 2018; 7: 74–110.

scholarly journals On the Spectrum of the Local $${\mathbb {P}}^2$$ Mirror Curve

2020 ◽  
Vol 21 (11) ◽  
pp. 3479-3497
Author(s):  
Rinat Kashaev ◽  
Sergey Sergeev
Keyword(s):  
Spectral Problem ◽  
Bethe Ansatz ◽  
Quantum Mechanical ◽  
Planck’S Constant ◽  
Transcendental Equations ◽  
Del Pezzo ◽  
Planck's Constant ◽  
Quantum Mechanical Operator ◽  
Mechanical Operator

Abstract We address the spectral problem of the formally normal quantum mechanical operator associated with the quantised mirror curve of the toric (almost) del Pezzo Calabi–Yau threefold called local $${\mathbb {P}}^2$$ P 2 in the case of complex values of Planck’s constant. We show that the problem can be approached in terms of the Bethe ansatz-type highly transcendental equations.

scholarly journals AN ENTROPIC PICTURE OF EMERGENT QUANTUM MECHANICS

2012 ◽  
Vol 09 (05) ◽  
pp. 1250048 ◽  
Author(s):  
D. ACOSTA ◽  
P. FERNÁNDEZ DE CÓRDOBA ◽  
J. M. ISIDRO ◽  
J. L. G. SANTANDER
Keyword(s):  
Quantum Mechanics ◽  
Quantum Mechanical ◽  
Entropy Flow ◽  
Planck’S Constant ◽  
Formal Derivation ◽  
Usual Quantum ◽  
Holographic Screen ◽  
Holographic Screens ◽  
Planck's Constant ◽  
Emergent Quantum Mechanics

Quantum mechanics emerges à la Verlinde from a foliation of ℝ3 by holographic screens, when regarding the latter as entropy reservoirs that a particle can exchange entropy with. This entropy is quantized in units of Boltzmann's constant kB. The holographic screens can be treated thermodynamically as stretched membranes. On that side of a holographic screen where spacetime has already emerged, the energy representation of thermodynamics gives rise to the usual quantum mechanics. A knowledge of the different surface densities of entropy flow across all screens is equivalent to a knowledge of the quantum-mechanical wavefunction on ℝ3. The entropy representation of thermodynamics, as applied to a screen, can be used to describe quantum mechanics in the absence of spacetime, that is, quantum mechanics beyond a holographic screen, where spacetime has not yet emerged. Our approach can be regarded as a formal derivation of Planck's constant ℏ from Boltzmann's constant kB.

scholarly journals Why space could be quantised on a different scale to matter

2021 ◽  
Author(s):  
Matthew J. Lake
Keyword(s):  
Degrees Of Freedom ◽  
Quantum Mechanical ◽  
Material Objects ◽  
Simple Argument ◽  
Planck’S Constant ◽  
Mechanical Effects ◽  
Quantum Mechanical Effects ◽  
Planck's Constant ◽  
The Universe

The scale of quantum mechanical effects in matter is set by Planck’s constant, \hbarℏ. This represents the quantisation scale for material objects. In this article, we present a simple argument why the quantisation scale for space, and hence for gravity, may not be equal to \hbarℏ. Indeed, assuming a single quantisation scale for both matter and geometry leads to the `worst prediction in physics’, namely, the huge difference between the observed and predicted vacuum energies. Conversely, assuming a different quantum of action for geometry, \beta \ll \hbarβ≪ℏ, allows us to recover the observed density of the Universe. Thus, by measuring its present-day expansion, we may in principle determine, empirically, the scale at which the geometric degrees of freedom should be quantised.

On the error of approximations in quantum mechanics. I. General theory

1965 ◽  
Vol 257 (1081) ◽  
pp. 309-326 ◽  
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Computer Applications ◽  
Quantum Mechanical ◽  
Quantum Mechanical Calculations ◽  
Density Matrices ◽  
Single Error

General formulas for estimating the errors in quantum-mechanical calculations are given in the formalism of density matrices. Some properties of the traces of matrices are used to simplify the estimating and to indicate a way of obtaining a better approximation. It is shown that the simultaneous correction of all the equations to be fulfilled leads in most cases to a faster convergence than the exact fulfilment of some of the equations and approximating stepwise to some of the others. The corrective formulas contain only direct operations of the matrices occurring and so they are advantageous in computer applications. In the last section a ‘subjective error’ definition is given and by taking into account the weight of the errors of the several equations a faster convergence and a single error quantity is obtained. Some special applications of the method will be published later.

N-Representable one-electron reduced density matrices reconstruction at non-zero temperatures

2021 ◽  
Vol 77 (5) ◽  
Author(s):  
Yoann Launay ◽  
Jean-Michel Gillet
Keyword(s):  
Density Matrix ◽  
Thermal Effects ◽  
Scattering Data ◽  
Temperature Dependent ◽  
Density Matrices ◽  
Reduced Density Matrices ◽  
Structure Factors ◽  
Reduced Density ◽  
Statistical Noise

This article retraces different methods that have been explored to account for the atomic thermal motion in the reconstruction of one-electron reduced density matrices from experimental X-ray structure factors (XSF) and directional Compton profiles (DCP). Attention has been paid to propose the simplest possible model, which obeys the necessary N-representability conditions, while accurately reproducing all available experimental data. The deconvolution of thermal effects makes it possible to obtain an experimental static density matrix, which can directly be compared with theoretical 1-RDM (reduced density matrix). It is found that above a 1% statistical noise level, the role played by Compton scattering data becomes negligible and no accurate 1-RDM is reachable. Since no thermal 1-RDM is available as a reference, the quality of an experimentally derived temperature-dependent matrix is difficult to assess. However, the accuracy of the obtained static 1-RDM, through the performance of the refined observables, is strong evidence that the Semi-Definite Programming method is robust and well adapted to the reconstruction of an experimental dynamical 1-RDM.

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