scholarly journals Short-Time Propagators and the Born–Jordan Quantization Rule

Entropy ◽  
2018 ◽  
Vol 20 (11) ◽  
pp. 869 ◽  
Author(s):  
Maurice de Gosson
Keyword(s):  
Quantum Mechanics ◽  
Quantization Scheme ◽  
Action Functional ◽  
Quantization Rule ◽  
Short Time ◽  
Quantization Rules

We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born–Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.

scholarly journals Short-Time Propagators and the Born--Jordan Quantization Rule

2018 ◽  
Author(s):  
Maurice A. de Gosson
Keyword(s):  
Quantum Mechanics ◽  
Quantization Scheme ◽  
Action Functional ◽  
Quantization Rule ◽  
Short Time ◽  
Quantization Rules

We have shown in previous work that the equivalence of the Heisenberg and Schrödinger pictures of quantum mechanics requires the use of the Born and Jordan quantization rules. In the present work we give further evidence that the Born--Jordan rule is the correct quantization scheme for quantum mechanics. For this purpose we use correct short-time approximations to the action functional, initially due to Makri and Miller, and show that these lead to the desired quantization of the classical Hamiltonian.

scholarly journals Schwarzian quantum mechanics as a Drinfeld-Sokolov reduction of BF theory

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Fridrich Valach ◽  
Donald R. Youmans
Keyword(s):  
Quantum Mechanics ◽  
Partition Function ◽  
Gauge Group ◽  
Path Integral ◽  
Coadjoint Orbits ◽  
Two Dimensional ◽  
Edge States ◽  
Class Constraint ◽  
Action Functional ◽  
Bf Theory

Abstract We give an interpretation of the holographic correspondence between two-dimensional BF theory on the punctured disk with gauge group PSL(2, ℝ) and Schwarzian quantum mechanics in terms of a Drinfeld-Sokolov reduction. The latter, in turn, is equivalent to the presence of certain edge states imposing a first class constraint on the model. The constrained path integral localizes over exceptional Virasoro coadjoint orbits. The reduced theory is governed by the Schwarzian action functional generating a Hamiltonian S1-action on the orbits. The partition function is given by a sum over topological sectors (corresponding to the exceptional orbits), each of which is computed by a formal Duistermaat-Heckman integral.

Energy and Momentum Eigenspectrum of the Hulthén-screened Cosine Kratzer Potential Using Proper Quantization Rule and SUSYQM Method

2021 ◽  
Author(s):  
Kaushal R Purohit ◽  
Rajendrasinh H PARMAR ◽  
Ajay Kumar Rai
Keyword(s):  
Quantum Mechanics ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Numerical Data ◽  
Time Dependent ◽  
Quantization Rule ◽  
Approximation Approach ◽  
Momentum Spectra ◽  
Energy And Momentum ◽  
Three Dimensional Space

Abstract Using the Qiang-Dong proper quantization rule (PQR) and the supersymmetric quantum mechanics approach, we obtained the eigenspectrum of the energy and momentum for time independent and time dependent Hulthen-screened cosine Kratzer potentials. For the suggested time independent Hulthen-screened cosine Kratzer potential, we solved the Schrodinger equation in D dimensions (HSCKP). The Feinberg-Horodecki equation for time-dependent Hulthen-screened cosine Kratzer potential was also solved (tHSCKP). To address the inverse square term in the time independent and time dependent equations, we employed the Greene-Aldrich approximation approach. We were able to extract time independent and time dependent potentials, as well as their accompanying energy and momentum spectra. In three-dimensional space, we estimated the rotational vibrational (RV) energy spectrum for many homodimers ($H_2, I_2, O_2$) and heterodimers ($MnH, ScN, LiH, HCl$). We also used the recently introduced formula approach to obtain the relevant eigen function. We also calculated momentum spectra for the dimers $MnH$ and $ScN$. The method is compared to prior methodologies for accuracy and validity using numerical data for heterodimer $LiH, HCl$ and homodimer $I_2, O_2,H_2$. The calculated energy and momentum spectra are tabulated and analysed.

scholarly journals Homogeneous Field and WKB Approximation in Deformed Quantum Mechanics with Minimal Length

2015 ◽  
Vol 2015 ◽  
pp. 1-15 ◽  
Author(s):  
Jun Tao ◽  
Peng Wang ◽  
Haitang Yang
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Statistical Physics ◽  
Turning Point ◽  
Minimal Length ◽  
Jacobi Method ◽  
Homogeneous Field ◽  
Quantization Rule ◽  
Connection Formula ◽  
Forbidden Region

In the framework of the deformed quantum mechanics with a minimal length, we consider the motion of a nonrelativistic particle in a homogeneous external field. We find the integral representation for the physically acceptable wave function in the position representation. Using the method of steepest descent, we obtain the asymptotic expansions of the wave function at large positive and negative arguments. We then employ the leading asymptotic expressions to derive the WKB connection formula, which proceeds from classically forbidden region to classically allowed one through a turning point. By the WKB connection formula, we prove the Bohr-Sommerfeld quantization rule up toOβ2. We also show that if the slope of the potential at a turning point is too steep, the WKB connection formula is no longer valid around the turning point. The effects of the minimal length on the classical motions are investigated using the Hamilton-Jacobi method. We also use the Bohr-Sommerfeld quantization to study statistical physics in deformed spaces with the minimal length.

scholarly journals Regular Bohr-Sommerfeld quantization rules for a h-pseudo-differential operator. The method of positive commutators

2016 ◽  
Vol Volume 23 - 2016 - Special... ◽  
Author(s):  
Abdelwaheb Ifa ◽  
Michel Rouleux
Keyword(s):  
Differential Operator ◽  
Gram Matrix ◽  
Pseudo Differential Operator ◽  
Quantization Rule ◽  
International Audience ◽  
Scalar Products ◽  
Quantization Rules

International audience We revisit in this Note the well known Bohr-Sommerfeld quantization rule (BS) for a 1-D Pseudo-differential self-adjoint Hamiltonian within the algebraic and microlocal framework of Helffer and Sjöstrand; BS holds precisely when the Gram matrix consisting of scalar products of some WKB solutions with respect to the " flux norm " is not invertible. Dans le cadre algébrique et microlocal élaboré par Helffer et Sjöstrand, on propose une ré-écriture de la règle de quantification de Bohr-Sommerfeld pour un opérateur auto-adjoint h-Pseudo-différentiel 1-D; elle s'exprime par la non-inversibilité de la matrice de Gram d'un couple de solutions WKB dans une base convenable, pour le produit scalaire associé à la " norme de flux " .

Chern–Simons gauge theory and symplectic quantum mechanics

2015 ◽  
Vol 93 (9) ◽  
pp. 971-973
Author(s):  
Lisa Jeffrey
Keyword(s):  
Quantum Mechanics ◽  
Gauge Theory ◽  
Partition Function ◽  
Partition Functions ◽  
Action Functional ◽  
Symplectic Action ◽  
Chern Simons

We describe the relation between the Chern–Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional. We also compute the semiclassical formulas for the partition functions obtained using the two different Lagrangians: the Chern–Simons functional and the symplectic action functional.

scholarly journals Fast-forward of adiabatic dynamics in quantum mechanics

2009 ◽  
Vol 466 (2116) ◽  
pp. 1135-1154 ◽  
Author(s):  
Shumpei Masuda ◽  
Katsuhiro Nakamura
Keyword(s):  
Quantum Mechanics ◽  
Wave Functions ◽  
Final Time ◽  
Final State ◽  
Previous Theory ◽  
Adiabatic Dynamics ◽  
Short Time ◽  
Driving Potential ◽  
Adiabatic State ◽  
The Ideal

We propose a method to accelerate adiabatic dynamics of wave functions (WFs) in quantum mechanics to obtain a final adiabatic state except for the spatially uniform phase in any desired short time. In our previous work, acceleration of the dynamics of WFs was shown to obtain the final state in any short time by applying driving potential. We develop the previous theory of fast-forward to derive a driving potential for the fast-forward of adiabatic dynamics. A typical example is the fast-forward of adiabatic transport of a WF, which is the ideal transport in the sense that a stationary WF is transported to an aimed position in any desired short time without leaving any disturbance at the final time of the fast-forward. As other important examples, we show accelerated manipulations of WFs, such as their splitting and squeezing. The theory is also applicable to macroscopic quantum mechanics described by the nonlinear Schrödinger equation.

scholarly journals Quantum mechanics, gravity and modified quantization relations

2015 ◽  
Vol 373 (2047) ◽  
pp. 20140244 ◽  
Author(s):  
Xavier Calmet
Keyword(s):  
Quantum Mechanics ◽  
Quantum Theory ◽  
Magnetic Moment ◽  
Anomalous Magnetic Moment ◽  
Scale Dependence ◽  
Energy Scale ◽  
Point Of View ◽  
Planck’S Constant ◽  
Phenomenological Point ◽  
Quantization Rules

In this paper, we investigate a possible energy scale dependence of the quantization rules and, in particular, from a phenomenological point of view, an energy scale dependence of an effective (reduced Planck’s constant). We set a bound on the deviation of the value of at the muon scale from its usual value using measurements of the anomalous magnetic moment of the muon. Assuming that inflation has taken place, we can conclude that nature is described by a quantum theory at least up to an energy scale of about 10 16  GeV.

scholarly journals Classical-quantum versus exact quantum results for a particle in a box

2012 ◽  
Vol 34 (2) ◽  
Author(s):  
Salvatore De Vincenzo
Keyword(s):  
Quantum Mechanics ◽  
Fourier Series ◽  
Classical Particle ◽  
Quantization Rule ◽  
One Dimensional ◽  
Exact Quantum ◽  
Classical Quantum

The problems of a free classical particle inside a one-dimensional box: (i) with impenetrable walls and (ii) with penetrable walls, were considered. For each problem, the classical amplitude and mechanical frequency of the T -th harmonic of the motion of the particle were identified from the Fourier series of the position function. After using the Bohr-Sommerfeld-Wilson quantization rule, the respective quantized amplitudes and frequencies (i.e., as a function of the quantum label n ) were obtained. Finally, the classical-quantum results were compared to those obtained from modern quantum mechanics, and a clear correspondence was observed in the limit of n » τ.

scholarly journals QUANTUM MECHANICS IN NONINERTIAL FRAMES WITH A MULTITEMPORAL QUANTIZATION SCHEME II: NONRELATIVISTIC PARTICLES

2006 ◽  
Vol 21 (19n20) ◽  
pp. 3917-3945 ◽  
Author(s):  
DAVID ALBA
Keyword(s):  
Quantum Mechanics ◽  
Bound States ◽  
Center Of Mass ◽  
Quantization Scheme ◽  
Unitary Evolution ◽  
Classical Level ◽  
Noninertial Effects ◽  
Special Cases ◽  
Relativistic Particles

The nonrelativistic version of the multitemporal quantization scheme of relativistic particles in a family of noninertial frames (see Ref. 1) is defined. At the classical level the description of a family of nonrigid noninertial frames, containing the standard rigidly linear accelerated and rotating ones, is given in the framework of parametrized Galilei theories. Then the multitemporal quantization, in which the gauge variables, describing the noninertial effects, are not quantized but considered as c-number generalized times, is applied to nonrelativistic particles. It is shown that with a suitable ordering there is unitary evolution in all times and that, after the separation of the center-of-mass, it is still possible to identify the inertial bound states. The few existing results of quantization in rigid noninertial frames are recovered as special cases.

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