scholarly journals REMARKS ON THE STATISTICAL ORIGIN OF THE GEOMETRICAL FORMULATION OF QUANTUM MECHANICS

2012 ◽  
Vol 09 (03) ◽  
pp. 1220001 ◽  
Author(s):  
MATHIEU MOLITOR
Keyword(s):  
Quantum Mechanics ◽  
Quantum Systems ◽  
Open Dense Subset ◽  
Kähler Structure ◽  
Mechanical Formalism ◽  
Finite Dimensional ◽  
Natural Geometry ◽  
Finite Set ◽  
Quantum Mechanical Formalism ◽  
Geometrical Formulation

A quantum system can be entirely described by the Kähler structure of the projective space [Formula: see text] associated to the Hilbert space [Formula: see text] of possible states; this is the so-called geometrical formulation of quantum mechanics. In this paper, we give an explicit link between the geometrical formulation (of finite dimensional quantum systems) and statistics through the natural geometry of the space [Formula: see text] of non-vanishing probabilities [Formula: see text] defined on a finite set En: = {x1, …, xn}. More precisely, we use the Fisher metric gF and the exponential connection ∇(1) (both being natural statistical objects living on [Formula: see text]) to construct, via the Dombrowski splitting theorem, a Kähler structure on [Formula: see text] which has the property that it induces the natural Kähler structure of a suitably chosen open dense subset of ℙ(ℂn). As a direct physical consequence, a significant part of the quantum mechanical formalism (in finite dimension) is encoded in the triple [Formula: see text].

ALTERNATIVE HAMILTONIAN DESCRIPTIONS FOR QUANTUM SYSTEMS AND NON-HERMITIAN OPERATORS WITH REAL SPECTRUM

2002 ◽  
Vol 17 (24) ◽  
pp. 1589-1599 ◽  
Author(s):  
FRANCO VENTRIGLIA
Keyword(s):  
Inverse Problem ◽  
Quantum Mechanics ◽  
Hilbert Spaces ◽  
Quantum Systems ◽  
Theoretical Physics ◽  
Inner Product ◽  
Real Spectrum ◽  
Standard Methods ◽  
Finite Dimensional

Many problems in theoretical physics are very frequently dealt with non-Hermitian operators. Recently there has been a lot of interest in non-Hermitian operators with real spectra. In this paper, by using the inverse problem for quantum systems, we show that, on finite-dimensional Hilbert spaces, all diagonalizable operators with a real spectrum can be made Hermitian with respect to a properly chosen inner product. In particular this allows the use of standard methods of quantum mechanics to analyze non-Hermitian operators with real spectra.

ALTERNATIVE HAMILTONIAN DESCRIPTION FOR QUANTUM SYSTEMS

1990 ◽  
Vol 05 (15) ◽  
pp. 1229-1234 ◽  
Author(s):  
B. A. DUBROVIN ◽  
G. MARMO ◽  
A. SIMONI
Keyword(s):  
Quantum Mechanics ◽  
Quantum Systems ◽  
Dimensional Case ◽  
Hamiltonian Description ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Kähler Structures ◽  
Time Invariant ◽  
Classical And Quantum Mechanics

The existence of time-invariant Kähler structures is analyzed in both Classical and Quantum Mechanics. In Quantum Mechanics, a family of such Kähler structures is found, in the finite-dimensional case it is proven that this family is complete.

scholarly journals A quantum vocal theory of sound

2020 ◽  
Vol 19 (9) ◽  
Author(s):  
Davide Rocchesso ◽  
Maria Mannone
Keyword(s):  
Quantum Mechanics ◽  
Temporal Evolution ◽  
Spin States ◽  
Vocal Production ◽  
Analysis And Design ◽  
Human Voice ◽  
Mechanical Formalism ◽  
Analysis And Synthesis ◽  
Quantum Mechanical Formalism ◽  
New Perspective

Abstract Concepts and formalism from acoustics are often used to exemplify quantum mechanics. Conversely, quantum mechanics could be used to achieve a new perspective on acoustics, as shown by Gabor studies. Here, we focus in particular on the study of human voice, considered as a probe to investigate the world of sounds. We present a theoretical framework that is based on observables of vocal production, and on some measurement apparati that can be used both for analysis and synthesis. In analogy to the description of spin states of a particle, the quantum-mechanical formalism is used to describe the relations between the fundamental states associated with phonetic labels such as phonation, turbulence, and supraglottal myoelastic vibrations. The intermingling of these states, and their temporal evolution, can still be interpreted in the Fourier/Gabor plane, and effective extractors can be implemented. The bases for a quantum vocal theory of sound, with implications in sound analysis and design, are presented.

scholarly journals Evidence for interactive common causes. Resuming the Cartwright-Hausman-Woodward debate

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Paul M. Näger
Keyword(s):  
Quantum Mechanics ◽  
Causal Model ◽  
Quantum Mechanical ◽  
Markov Condition ◽  
Causal Modelling ◽  
Mechanical Formalism ◽  
Common Causes ◽  
Quantum Mechanical Formalism ◽  
Causal Markov Condition ◽  
Quantum Phenomena

AbstractThe most serious candidates for common causes that fail to screen off (‘interactive common causes’, ICCs) and thus violate the causal Markov condition (CMC) refer to quantum phenomena. In her seminal debate with Hausman and Woodward, Cartwright early on focussed on unfortunate non-quantum examples. Especially, Hausman and Woodward’s redescriptions of quantum cases saving the CMC remain unchallenged. This paper takes up this lose end of the discussion and aims to resolve the debate in favour of Cartwright’s position. It systematically considers redescriptions of ICC structures, including those by Hausman and Woodward, and explains why these are inappropriate, when quantum mechanics (in an objective collapse interpretation) is true. It first shows that all cases of purported quantum ICCs are cases of entanglement and then, using the tools of causal modelling, it provides an analysis of the quantum mechanical formalism for the case that the collapse of entangled systems is best described as a causal model with an ICC.

Relativistic Multiple Scattering Theory

1991 ◽  
Vol 253 ◽  
Author(s):  
B. L. Gyorffy
Keyword(s):  
Quantum Mechanics ◽  
Degrees Of Freedom ◽  
Relativistic Quantum ◽  
Schr6dinger Equation ◽  
Relativistic Effects ◽  
Relativistic Quantum Mechanics ◽  
Absolute Minimum ◽  
Crystalline Anisotropy ◽  
Symmetry Properties ◽  
Finite Set

The symmetry properties of the Dirac equation, which describes electrons in relativistic quantum mechanics, is rather different from that of the corresponding Schr6dinger equation. Consequently, even when the velocity of light, c, is much larger than the velocity of an electron Vk, with wave vector, k, relativistic effects may be important. For instance, while the exchange interaction is isotropic in non-relativistic quantum mechanics the coupling between spin and orbital degrees of freedom in relativistic quantum mechanics implies that the band structure of a spin polarized metal depends on the orientation of its magnetization with respect to the crystal axis. As a consequence there is a finite set of degenerate directions for which the total energy of the electrons is an absolute minimum. Evidently, the above effect is the principle mechanism of the magneto crystalline anisotropy [1]. The following session will focus on this and other qualitatively new relativistic effects, such as dichroism at x-ray frequencies [2] or Fano effects in photo-emission from non-polarized solids [3].

scholarly journals Wigner's Problem and Alternative Commutation Relations for Quantum Mechanics

1997 ◽  
Vol 11 (10) ◽  
pp. 1281-1296 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
F. Zaccaria ◽  
E. C. G. Sudarshan
Keyword(s):  
Quantum Mechanics ◽  
Vector Field ◽  
Equations Of Motion ◽  
Quantum Systems ◽  
Heisenberg Picture ◽  
Commutation Relations

It is shown that for quantum systems the vector field associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrödinger and Heisenberg picture. We illustrate these ambiguities in terms of simple examples.

scholarly journals Using quantum computing to create efficient algorithms

2021 ◽  
Vol 2056 (1) ◽  
pp. 012059
Author(s):  
I N Balaba ◽  
G S Deryabina ◽  
I A Pinchuk ◽  
I V Sergeev ◽  
S B Zabelina
Keyword(s):  
Quantum Mechanics ◽  
Quantum Computing ◽  
Computational Model ◽  
Exponential Growth ◽  
Quantum Algorithms ◽  
Quantum Systems ◽  
Efficient Algorithms ◽  
Historical Overview ◽  
Computing Model ◽  
Computational Strategy

Abstract The article presents a historical overview of the development of the mathematical idea of a quantum computing model - a new computational strategy based on the postulates of quantum mechanics and having advantages over the traditional computational model based on the Turing machine; clarified the features of the operation of multi-qubit quantum systems, which ensure the creation of efficient algorithms; the principles of quantum computing are outlined and a number of efficient quantum algorithms are described that allow solving the problem of exponential growth of the complexity of certain problems.

scholarly journals On the Generation of Random Ensembles of Qubits and Qutrits Computing Separability Probabilities for Fixed Rank States

2018 ◽  
Vol 173 ◽  
pp. 02010 ◽  
Author(s):  
Arsen Khvedelidze ◽  
Ilya Rogojin
Keyword(s):  
Quantum Systems ◽  
Separable State ◽  
Mixed States ◽  
Finite Dimensional ◽  
Fixed Rank

The generation of random mixed states is discussed, aiming for the computation of probabilistic characteristics of composite finite dimensional quantum systems. In particular, we consider the generation of random Hilbert-Schmidt and Bures ensembles of qubit and qutrit pairs and compute the corresponding probabilities to find a separable state among the states of a fixed rank.

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