scholarly journals ADELIC HARMONIC OSCILLATOR

1995 ◽  
Vol 10 (16) ◽  
pp. 2349-2365 ◽  
Author(s):  
BRANKO DRAGOVICH
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Functional Relation ◽  
Vacuum State ◽  
One Dimensional ◽  
High Energies ◽  
The Riemann Zeta Function ◽  
Adelic Quantum Mechanics ◽  
Riemann Zeta ◽  
Very High

Using the Weyl quantization we formulate one-dimensional adelic quantum mechanics, which unifies and treats ordinary and p-adic quantum mechanics on an equal footing. As an illustration the corresponding harmonic oscillator is considered. It is a simple, exact and instructive adelic model. Eigenstates are Schwartz-Bruhat functions. The Mellin transform of the simplest vacuum state leads to the well-known functional relation for the Riemann zeta function. Some expectation values are calculated. The existence of adelic matter at very high energies is suggested.

The Riemann Zeta Function and the Inverted Harmonic Oscillator

1997 ◽  
Vol 254 (1) ◽  
pp. 25-40 ◽  
Author(s):  
R.K. Bhaduri ◽  
Avinash Khare ◽  
S.M. Reimann ◽  
E.L. Tomusiak
Keyword(s):  
Harmonic Oscillator ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
The Riemann Zeta Function ◽  
Riemann Zeta

scholarly journals NEW FEATURES IN SUPERSYMMETRY BREAKDOWN IN QUANTUM MECHANICS

1996 ◽  
Vol 11 (19) ◽  
pp. 1563-1567 ◽  
Author(s):  
BORIS F. SAMSONOV
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Mechanical Model ◽  
State Energy ◽  
Vacuum State ◽  
Quantum Mechanical ◽  
Oscillator Potential ◽  
Harmonic Oscillator Potential ◽  
Quantum Mechanical Model ◽  
Higher Derivative

The supersymmetric quantum mechanical model based on higher-derivative supercharge operators possessing unbroken supersymmetry and discrete energies below the vacuum state energy is described. As an example harmonic oscillator potential is considered.

Quantum mechanics on (anti)-de Sitter background II: Ramsauer–Townsend effect and WKB method

2018 ◽  
Vol 33 (26) ◽  
pp. 1850150 ◽  
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Energy Level ◽  
Harmonic Oscillator ◽  
Wkb Method ◽  
De Sitter ◽  
Quantum Harmonic Oscillator ◽  
One Dimensional ◽  
Sitter Background ◽  
The One

Based on the one-dimensional quantum mechanics on (anti)-de Sitter background [W. S. Chung and H. Hassanabadi, Mod. Phys. Lett. A 32, 26 (2107)], we discuss the Ramsauer–Townsend effect. We also formulate the WKB method for the quantum mechanics on (anti)-de Sitter background to discuss the energy level of the quantum harmonic oscillator and quantum bouncer.

scholarly journals Non-Hermitian inverted harmonic oscillator-type Hamiltonians generated from supersymmetry with reflections

2019 ◽  
Vol 34 (04) ◽  
pp. 1950028
Author(s):  
R. D. Mota ◽  
D. Ojeda-Guillén ◽  
M. Salazar-Ramírez ◽  
V. D. Granados
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Supersymmetric Quantum Mechanics ◽  
Type Model ◽  
One Dimensional ◽  
The Third ◽  
Derivative Formula ◽  
Supersymmetric Models ◽  
Oscillator Type ◽  
The Creation

By modifying and generalizing known supersymmetric models, we are able to find four different sets of one-dimensional Hamiltonians for the inverted harmonic oscillator. The first set of Hamiltonians is derived by extending the supersymmetric quantum mechanics with reflections to non-Hermitian supercharges. The second set is obtained by generalizing the supersymmetric quantum mechanics valid for non-Hermitian supercharges with the Dunkl derivative instead of [Formula: see text]. Also, by changing the derivative [Formula: see text] by the Dunkl derivative in the creation and annihilation-type operators of the standard inverted harmonic oscillator [Formula: see text], we generate the third set of Hamiltonians. The fourth set of Hamiltonians emerges by allowing a parameter of the supersymmetric two-body Calogero-type model to take imaginary values. The eigensolutions of definite parity for each set of Hamiltonians are given.

THE RIEMANN HYPOTHESIS IS A CONSEQUENCE OF $\mathcal{CT}$-INVARIANT QUANTUM MECHANICS

2008 ◽  
Vol 05 (01) ◽  
pp. 17-32
Author(s):  
CARLOS CASTRO PERELMAN
Keyword(s):  
Quantum Mechanics ◽  
Riemann Hypothesis ◽  
Mellin Transform ◽  
Theta Series ◽  
Infinite Family ◽  
Continuous Family ◽  
Nontrivial Zeros ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Invariant Quantum

The Riemann's hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form sn = 1/2 + iλn. By constructing a continuous family of scaling-like operators involving the Gauss–Jacobi theta series and by invoking a novel [Formula: see text]-invariant Quantum Mechanics, involving a judicious charge conjugation [Formula: see text] and time reversal [Formula: see text] operation, we show why the Riemann Hypothesis is true. An infinite family of theta series and their Mellin transform leads to the same conclusions.

One-dimensional quantum mechanics with Dunkl derivative

2019 ◽  
Vol 34 (24) ◽  
pp. 1950190
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Scalar Product ◽  
Momentum Operator ◽  
Inner Product ◽  
One Dimensional ◽  
Coordinate Representation ◽  
The One ◽  
Infinite Wall

In this paper, we consider the quantum mechanics with Dunkl derivative. We use the Dunkl derivative to obtain the coordinate representation of the momentum operator and Hamiltonian. We introduce the scalar product to find that the momentum is Hermitian under this inner product. We study the one-dimensional box problem (the spin-less particle with mass m confined to the one-dimensional infinite wall). Finally, we discuss the harmonic oscillator problem.

scholarly journals The Generalized OTOC from Supersymmetric Quantum Mechanics—Study of Random Fluctuations from Eigenstate Representation of Correlation Functions

Symmetry ◽  
2020 ◽  
Vol 13 (1) ◽  
pp. 44
Author(s):  
Kaushik Y. Bhagat ◽  
Baibhab Bose ◽  
Sayantan Choudhury ◽  
Satyaki Chowdhury ◽  
Rathindra N. Das ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Quantum Statistical Mechanics ◽  
Supersymmetric Quantum Mechanics ◽  
The Other ◽  
Potential Well ◽  
One Dimensional ◽  
Quantum Randomness ◽  
The One ◽  
The Impact

The concept of the out-of-time-ordered correlation (OTOC) function is treated as a very strong theoretical probe of quantum randomness, using which one can study both chaotic and non-chaotic phenomena in the context of quantum statistical mechanics. In this paper, we define a general class of OTOC, which can perfectly capture quantum randomness phenomena in a better way. Further, we demonstrate an equivalent formalism of computation using a general time-independent Hamiltonian having well-defined eigenstate representation for integrable Supersymmetric quantum systems. We found that one needs to consider two new correlators apart from the usual one to have a complete quantum description. To visualize the impact of the given formalism, we consider the two well-known models, viz. Harmonic Oscillator and one-dimensional potential well within the framework of Supersymmetry. For the Harmonic Oscillator case, we obtain similar periodic time dependence but dissimilar parameter dependences compared to the results obtained from both micro-canonical and canonical ensembles in quantum mechanics without Supersymmetry. On the other hand, for the One-Dimensional Potential Well problem, we found significantly different time scales and the other parameter dependence compared to the results obtained from non-Supersymmetric quantum mechanics. Finally, to establish the consistency of the prescribed formalism in the classical limit, we demonstrate the phase space averaged version of the classical version of OTOCs from a model-independent Hamiltonian, along with the previously mentioned well-cited models.

scholarly journals Riemann Hypothesis and Random Walks: The Zeta Case

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2014
Author(s):  
André LeClair
Keyword(s):  
Zeta Function ◽  
Riemann Hypothesis ◽  
Critical Line ◽  
Dirichlet Characters ◽  
The Riemann Zeta Function ◽  
Point Correlation ◽  
Point Correlation Function ◽  
Riemann Zeta ◽  
The Right ◽  
Very High

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

scholarly journals Adelic Path Integrals for Quadratic Lagrangians

2003 ◽  
Vol 06 (02) ◽  
pp. 179-195 ◽  
Author(s):  
Goran S. Djordjević ◽  
Branko Dragovich ◽  
Ljubiša Nešić
Keyword(s):  
Quantum Mechanics ◽  
General Formula ◽  
Path Integral ◽  
Path Integrals ◽  
Number Fields ◽  
One Dimensional ◽  
Adelic Quantum Mechanics

Feynman's path integral in adelic quantum mechanics is considered. The propagator [Formula: see text] for one-dimensional adelic systems with quadratic Lagrangians is analytically evaluated. Obtained exact general formula has the form which is invariant under interchange of the number fields ℝ and ℚp.

Quantum mechanics on (anti)-de Sitter background

2017 ◽  
Vol 32 (26) ◽  
pp. 1750138 ◽  
Author(s):  
Won Sang Chung ◽  
Hassan Hassanabadi
Keyword(s):  
Quantum Mechanics ◽  
Harmonic Oscillator ◽  
Uncertainty Principle ◽  
De Sitter ◽  
One Dimensional ◽  
Sitter Background ◽  
Extended Uncertainty

In this paper, the quantum mechanics on the (anti) de Sitter background is investigated. the extended uncertainty principle and the deformed calculus are discussed for the quantum mechanics on the (anti)-de Sitter background. As examples one-dimensional box problem and one-dimensional harmonic oscillator problem are discussed.

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