On the symplectically invariant variational principle and generating functions

1990 ◽  
Vol 431 (1883) ◽  
pp. 403-417 ◽  
Keyword(s):  
Quantum Mechanics ◽  
Variational Principle ◽  
Generating Function ◽  
Generating Functions ◽  
Geometric Approach ◽  
Canonical Transformations ◽  
Weyl Transform ◽  
Analogous Relation ◽  
Definition Of ◽  
Symplectic Invariance

The generating function for canonical transformations derived by Marinov has the important property of symplectic invariance (i. e. under linear canonical transformations). However, a more geometric approach to the rederivation of this function from the variational principle reveals that it is not free from caustic singularities after all. These singularities can be avoided without breaking the symplectic invariance by the definition of a complementary generating function bearing an analogous relation to the Woodward ambiguity function in telecommunications theory as that tying Marinov’s function to the Wigner function and the Weyl transform in quantum mechanics. Marinov’s function is specially apt to describe canonical transformations close to the identity, but breaks down for reflections through a point in phase space, easily described by the new generating function.

scholarly journals CANONICAL TRANSFORMATIONS IN THREE-DIMENSIONAL PHASE-SPACE

2009 ◽  
Vol 24 (25n26) ◽  
pp. 4769-4788 ◽  
Author(s):  
TEKİN DERELİ ◽  
ADNAN TEĞMEN ◽  
TUĞRUL HAKİOĞLU
Keyword(s):  
Phase Space ◽  
Canonical Transformation ◽  
Generating Functions ◽  
General Framework ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Canonical Transformations ◽  
Dimensional Phase Space ◽  
Nambu Bracket ◽  
Definition Of

Canonical transformation in a three-dimensional phase-space endowed with Nambu bracket is discussed in a general framework. Definition of the canonical transformations is constructed based on canonoid transformations. It is shown that generating functions, transformed Hamilton functions and the transformation itself for given generating functions can be determined by solving Pfaffian differential equations corresponding to that quantities. Types of the generating functions are introduced and all of them are listed. Infinitesimal canonical transformations are also discussed. Finally, we show that the decomposition of canonical transformations is also possible in three-dimensional phase space as in the usual two-dimensional one.

scholarly journals QUANTUM CANONICAL TRANSFORMATIONS IN WEYL–WIGNER–GROENEWOLD–MOYAL FORMALISM

2009 ◽  
Vol 24 (24) ◽  
pp. 4573-4587 ◽  
Author(s):  
TEKİN DERELİ ◽  
TUĞRUL HAKİOĞLU ◽  
ADNAN TEĞMEN
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Generating Function ◽  
Canonical Transformation ◽  
Star Product ◽  
Product Form ◽  
Canonical Transformations ◽  
Exponential Form ◽  
Alternative Approach ◽  
Point Transformations

A conjecture in quantum mechanics states that any quantum canonical transformation can decompose into a sequence of three basic canonical transformations; gauge, point and interchange of coordinates and momenta. It is shown that if one attempts to construct the three basic transformations in star-product form, while gauge and point transformations are immediate in star-exponential form, interchange has no correspondent, but it is possible in an ordinary exponential form. As an alternative approach, it is shown that all three basic transformations can be constructed in the ordinary exponential form and that in some cases this approach provides more useful tools than the star-exponential form in finding the generating function for given canonical transformation or vice versa. It is also shown that transforms of c-number phase space functions under linear–nonlinear canonical transformations and intertwining method can be treated within this argument.

scholarly journals Canonical transformations and the Hamilton-Jacobi theory in quantum mechanics

1999 ◽  
Vol 77 (6) ◽  
pp. 411-425 ◽  
Author(s):  
J -H Kim ◽  
H -W Lee
Keyword(s):  
Quantum Mechanics ◽  
Phase Space ◽  
Generating Functions ◽  
Jacobi Equation ◽  
Classical Case ◽  
Time Dependent ◽  
Canonical Transformations ◽  
Hamilton Jacobi Equation ◽  
Number Form

Canonical transformations using the idea of quantum generating functions are applied to construct a quantum Hamilton-Jacobi theory, based on the analogy with the classical case. An operator and a c-number form of the time-dependent quantum Hamilton-Jacobi equation are derived and used to find dynamical solutions of quantum problems. The phase-space picture of quantum mechanics is discussed in connection with the present theory.PACS Nos.: 03.65-w, 03.65Ca, 03.65Ge

scholarly journals Induction Models on $\mathbb{N}$

10.29007/kvp3 ◽  
2020 ◽  
Author(s):  
A. Dileep ◽  
Kuldeep S. Meel ◽  
Ammar F. Sabili
Keyword(s):  
Computer Science ◽  
Generating Function ◽  
Generating Functions ◽  
Mathematical Induction ◽  
Base Case ◽  
And Mathematics ◽  
Definition Of ◽  
Induction Model

Mathematical induction is a fundamental tool in computer science and mathematics. Henkin [12] initiated the study of formalization of mathematical induction restricted to the setting when the base case B is set to singleton set containing 0 and a unary generating function S. The usage of mathematical induction often involves wider set of base cases and k−ary generating functions with different structural restrictions. While subsequent studies have shown several Induction Models to be equivalent, there does not exist precise logical characterization of reduction and equivalence among different Induction Models. In this paper, we generalize the definition of Induction Model and demonstrate existence and construction of S for given B and vice versa. We then provide a formal characterization of the reduction among different Induction Models that can allow proofs in one Induction Models to be expressed as proofs in another Induction Models. The notion of reduction allows us to capture equivalence among Induction Models.

Dualism QM and GR

2019 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Quantum Mechanics ◽  
General Theory ◽  
Noncommutative Geometry ◽  
Quantum Tunneling ◽  
Closed Surface ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Surface Integral ◽  
Definition Of

Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral. Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.

System of thermodynamic quantities in the McMillan-Mayer solution theory

1985 ◽  
Vol 50 (4) ◽  
pp. 791-798 ◽  
Author(s):  
Vilém Kodýtek
Keyword(s):  
Free Energy ◽  
Generating Function ◽  
Simple Form ◽  
Unit Volume ◽  
Thermodynamic Quantities ◽  
Solution Theory ◽  
Pure Solvent ◽  
Second Derivatives ◽  
Definition Of ◽  
Derivatives Of

The McMillan-Mayer (MM) free energy per unit volume of solution AMM, is employed as a generating function of the MM system of thermodynamic quantities for solutions in the state of osmotic equilibrium with pure solvent. This system can be defined by replacing the quantities G, T, P, and m in the definition of the Lewis-Randall (LR) system by AMM, T, P0, and c (P0 being the pure solvent pressure). Following this way the LR to MM conversion relations for the first derivatives of the free energy are obtained in a simple form. New relations are derived for its second derivatives.

Poisson Brackets & Angular Momentum

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Generating Functions ◽  
First Integrals ◽  
Lagrangian Mechanics ◽  
Poisson Brackets ◽  
Coordinate Transformations ◽  
Canonical Transformations ◽  
Gauge Transformations ◽  
The Third ◽  
Point Transformations ◽  
Canonical Coordinate

This chapter discusses canonical transformations and gauge transformations and is divided into three sections. In the first section, canonical coordinate transformations are introduced to the reader through generating functions as the extension of point transformations used in Lagrangian mechanics, with the harmonic oscillator being used as an example of a canonical transformation. In the second section, gauge theory is discussed in the canonical framework and compared to the Lagrangian case. Action-angle variables, direct conditions, symplectomorphisms, holomorphic variables, integrable systems and first integrals are examined. The third section looks at infinitesimal canonical transformations resulting from functions on phase space. Ostrogradsky equations in the canonical setting are also detailed.

scholarly journals A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1161
Author(s):  
Hari Mohan Srivastava ◽  
Sama Arjika
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Additional Parameter ◽  
Physical Sciences ◽  
General Family ◽  
Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

scholarly journals Generating Functions for Certain Weighted Cranks

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Shreejit Bandyopadhyay ◽  
Ae Yee
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Colored Partitions ◽  
Unimodal Sequences

Recently, George Beck posed many interesting partition problems considering the number of ones in partitions. In this paper, we first consider the crank generating function weighted by the number of ones and obtain analytic formulas for this weighted crank function under conditions of the crank being less than or equal to some specific integer. We connect these cumulative and point crank functions to the generating functions of partitions with certain sizes of Durfee rectangles. We then consider a generalization of the crank for $k$-colored partitions, which was first introduced by Fu and Tang, and investigate the corresponding generating function for this crank weighted by the number of parts in the first subpartition of a $k$-colored partition. We show that the cumulative generating functions are the same as the generating functions for certain unimodal sequences.

scholarly journals The effective mass problem for the Landau-Pekar equations

2021 ◽  
Author(s):  
Dario Feliciangeli ◽  
Simone Rademacher ◽  
Robert Seiringer
Keyword(s):  
Variational Principle ◽  
Effective Mass ◽  
Initial Velocity ◽  
Energy Functional ◽  
Mass Problem ◽  
Definition Of

Abstract We provide a definition of the effective mass for the classical polaron described by the Landau-Pekar equations. It is based on a novel variational principle, minimizing the energy functional over states with given (initial) velocity. The resulting formula for the polaron's effective mass agrees with the prediction by Landau and Pekar.

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