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

Hamiltonian Mechanics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Angular Momentum ◽  
Jacobi Equation ◽  
Modern Theory ◽  
Hamiltonian Mechanics ◽  
Lagrangian Mechanics ◽  
Canonical Transformations ◽  
Canonical Variables ◽  
Light Waves ◽  
Hamilton Jacobi Equation ◽  
Common Action

Hamilton’s genius was to understand what were the true variables of mechanics (the “p − q,” conjugate coordinates, or canonical variables), and this led to Hamilton’s Mechanics which could obtain qualitative answers to a wider ranger of problems than Lagrangian Mechanics. It is explained how Hamilton’s canonical equations arise, why the Hamiltonian is the “central conception of all modern theory” (quote of Schrödinger’s), what the “p − q” variables are, and what phase space is. It is also explained how the famous conservation theorems arise (for energy, linear momentum, and angular momentum), and the connection with symmetry. The Hamilton-Jacobi Equation is derived using infinitesimal canonical transformations (ICTs), and predicts wavefronts of “common action” spreading out in (configuration) space. An analogy can be made with geometrical optics and Huygen’s Principle for the spreading out of light waves. It is shown how Hamilton’s Mechanics can lead into quantum mechanics.

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 The Equivalence Postulate of Quantum Mechanics, Dark Energy, and the Intrinsic Curvature of Elementary Particles

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Alon E. Faraggi
Keyword(s):  
Quantum Mechanics ◽  
Dark Energy ◽  
Schrödinger Equation ◽  
Jacobi Equation ◽  
Schrodinger Equation ◽  
Axiomatic Approach ◽  
Quantum Potential ◽  
Curvature Term ◽  
Cocycle Condition ◽  
Hamilton Jacobi Equation

The equivalence postulate of quantum mechanics offers an axiomatic approach to quantum field theories and quantum gravity. The equivalence hypothesis can be viewed as adaptation of the classical Hamilton-Jacobi formalism to quantum mechanics. The construction reveals two key identities that underlie the formalism in Euclidean or Minkowski spaces. The first is a cocycle condition, which is invariant underD-dimensional Möbius transformations with Euclidean or Minkowski metrics. The second is a quadratic identity which is a representation of theD-dimensional quantum Hamilton-Jacobi equation. In this approach, the solutions of the associated Schrödinger equation are used to solve the nonlinear quantum Hamilton-Jacobi equation. A basic property of the construction is that the two solutions of the corresponding Schrödinger equation must be retained. The quantum potential, which arises in the formalism, can be interpreted as a curvature term. The author proposes that the quantum potential, which is always nontrivial and is an intrinsic energy term characterising a particle, can be interpreted as dark energy. Numerical estimates of its magnitude show that it is extremely suppressed. In the multiparticle case the quantum potential, as well as the mass, is cumulative.

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.

Quantum Mechanics and the Relativistic Hamilton-Jacobi Equation

1964 ◽  
Vol 133 (6B) ◽  
pp. B1622-B1624 ◽  
Author(s):  
Lloyd Motz ◽  
Adolph Selzer
Keyword(s):  
Quantum Mechanics ◽  
Jacobi Equation ◽  
Hamilton Jacobi Equation

scholarly journals Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

2021 ◽  
Author(s):  
Piermarco Cannarsa ◽  
Wei Cheng ◽  
Albert Fathi
Keyword(s):  
Riemannian Manifold ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Jacobi Equation ◽  
Compact Manifold ◽  
Complete Riemannian Manifold ◽  
Time Dependent ◽  
Jacobi Equations ◽  
Uniformly Continuous ◽  
Hamilton Jacobi Equation

AbstractIf $U:[0,+\infty [\times M$ U : [ 0 , + ∞ [ × M is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$ \partial _{t}U+ H(x,\partial _{x}U)=0, $$ ∂ t U + H ( x , ∂ x U ) = 0 , where $M$ M is a not necessarily compact manifold, and $H$ H is a Tonelli Hamiltonian, we prove the set $\Sigma (U)$ Σ ( U ) , of points in $]0,+\infty [\times M$ ] 0 , + ∞ [ × M where $U$ U is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma (U)$ Σ ( U ) . We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

scholarly journals Hamilton-Jacobi Equation of Time Dependent Hamiltonians

2020 ◽  
Vol 5 (1-2) ◽  
pp. 09-15
Author(s):  
Anoud K. Fuqara ◽  
Amer D. Al-Oqali ◽  
Khaled I. Nawafleh
Keyword(s):  
Hamiltonian Systems ◽  
Jacobi Equation ◽  
Classical Mechanics ◽  
Time Dependent ◽  
Calculus Of Variation ◽  
Jacobi Formalism ◽  
Equation Of Time ◽  
Hamilton Jacobi Equation

In this work, we apply the geometric Hamilton-Jacobi theory to obtain solution of Hamiltonian systems in classical mechanics that are either compatible with two structures: the first structure plays a central role in the theory of time- dependent Hamiltonians, whilst the second is used to treat classical Hamiltonians including dissipation terms. It is proved that the generalization of problems from the calculus of variation methods in the nonstationary case can be obtained naturally in Hamilton-Jacobi formalism.

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