Topic

# Canonical transformation

About: Canonical transformation is a(n) research topic. Over the lifetime, 1854 publication(s) have been published within this topic receiving 38019 citation(s).

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TL;DR: In this paper, a canonical transformation on the Dirac Hamiltonian for a free particle is obtained in which positive and negative energy states are separately represented by two-component wave functions.

Abstract: By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and $c$. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.

1,604 citations

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TL;DR: In this article, the behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations, and the results are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.

Abstract: The behavior of the electrons in a dense electron gas is analyzed quantum-mechanically by a series of canonical transformations. The usual Hamiltonian corresponding to a system of individual electrons with Coulomb interactions is first re-expressed in such a way that the long-range part of the Coulomb interactions between the electrons is described in terms of collective fields, representing organized "plasma" oscillation of the system as a whole. The Hamiltonian then describes these collective fields plus a set of individual electrons which interact with the collective fields and with one another via short-range screened Coulomb interactions. There is, in addition, a set of subsidiary conditions on the system wave function which relate the field and particle variables. The field-particle interaction is eliminated to a high degree of approximation by a further canonical transformation to a new representation in which the Hamiltonian describes independent collective fields, with ${n}^{\ensuremath{'}}$ degrees of freedom, plus the system of electrons interacting via screened Coulomb forces with a range of the order of the inter electronic distance. The new subsidiary conditions act only on the electronic wave functions; they strongly inhibit long wavelength electronic density fluctuations and act to reduce the number of individual electronic degrees of freedom by ${n}^{\ensuremath{'}}$. The general properties of this system are discussed, and the methods and results obtained are related to the classical density fluctuation approach and Tomonaga's one-dimensional treatment of the degenerate Fermi gas.

1,290 citations

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31 Jan 1971

TL;DR: The three-body problem was studied in this paper, where Covarinace of Lagarangian Derivatives and Canonical Transformation were applied to the problem of estimating the perimeter and the velocity of the system.

Abstract: The Three-Body Problem: Covarinace of Lagarangian Derivatives.- Canonical Transformation.- The Hamilton-Jacobi Equation.- The Cauchy-Existence Theorem.- The n-Body Poblem.- Collision.- The Regularizing Transformation.- Application to the Three-Bdy Problem.- An Estimate of the Perimeter.- An Estimate of the Velocity.- Sundman's Theorem.- Triple Collision.- Triple-Collision Orbits.- Periodic Solutions: The Solutions of Lagrange.- Eigenvalues.- An Existence Theorem.- The Convergence Proof.- An Application to the Solution of Lagrange.- Hill's Problem.- A Generalization of Hill's Problem.- The Continuation Method.- The Fixed-Point Theorem.- Area-Preserving Analytic Transformations.- The Birkhoff Fixed-Point Theorem.- Stability: The Function-Theoretic Center Problem.- The Convergence Proof.- The Poincare Center Problem.- The Theorem of Liapunov.- The Theorem of Dirichlet.- The Normal Form of Hamiltonian Systems.- Area-Preserving Transformations.- Existence of Invariant Curves.- Proof of Lemma.- Application to the Stability Problem.- Stability of Equilibrium Solutions.- Quasi-Periodic Motion and Systems of Several Degrees of Freedom.- The Recurrence Theorem.

1,049 citations

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TL;DR: In this paper, a variational technique was developed to investigate the low-lying energy levels of a conduction electron in a polar crystal, which is equivalent to a simple canonical transformation, and the use of this transformation enables us to obtain the wave functions and energy levels quite simply.

Abstract: A variational technique is developed to investigate the low-lying energy levels of a conduction electron in a polar crystal. Because of the strong interaction between the electron and the longitudinal optical mode of the lattice vibrations, perturbation-theoretic methods are inapplicable. Our variational technique, which is closely related to the "intermediate coupling" method introduced by Tomonaga, is equivalent to a simple canonical transformation. The use of this transformation enables us to obtain the wave functions and energy levels quite simply. Because the recoil of the electron introduces a correlation between the emission of successive virtual phonons by the electron, our approximation, in which this correlation is neglected, breaks down for very strong electron-phonon coupling. The validity of our approximation is investigated and corrections are found to be small for coupling strengths occurring in typical polar crystals.

823 citations

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TL;DR: In this article, a variational technique was developed to investigate the low-lying energy levels of a conduction electron in a polar crystal, which is equivalent to a simple canonical transformation, and the use of this transformation enables us to obtain the wave functions and energy levels quite simply.

Abstract: A variational technique is developed to investigate the low-lying energy levels of a conduction electron in a polar crystal. Because of the strong interaction between the electron and the longitudinal optical mode of the lattice vibrations, perturbation-theoretic methods are inapplicable. Our variational technique, which is closely related to the "intermediate coupling" method introduced by Tomonaga, is equivalent to a simple canonical transformation. The use of this transformation enables us to obtain the wave functions and energy levels quite simply. Because the recoil of the electron introduces a correlation between the emission of successive virtual phonons by the electron, our approximation, in which this correlation is neglected, breaks down for very strong electron-phonon coupling. The validity of our approximation is investigated and corrections are found to be small for coupling strengths occurring in typical polar crystals.

720 citations