Search results for "Canonical coordinates"
showing 5 items of 5 documents
Heisenberg Uncertainty Relation in Quantum Liouville Equation
2009
We consider the quantum Liouville equation and give a characterization of the solutions which satisfy the Heisenberg uncertainty relation. We analyze three cases. Initially we consider a particular solution of the quantum Liouville equation: the Wigner transformf(x,v,t) of a generic solutionψ(x;t) of the Schrödinger equation. We give a representation ofψ(x,t) by the Hermite functions. We show that the values of the variances ofxandvcalculated by using the Wigner functionf(x,v,t) coincide, respectively, with the variances of position operatorX^and conjugate momentum operatorP^obtained using the wave functionψ(x,t). Then we consider the Fourier transform of the density matrixρ(z,y,t) =ψ∗(z,t)…
Canonical Adiabatic Theory
2001
In the present chapter we are concerned with systems, the change of which—with the exception of a single degree of freedom—should proceed slowly. (Compare the pertinent remarks about \(\varepsilon\) as slow parameter in Chap. 7) Accordingly, the Hamiltonian reads: $$\displaystyle{ H = H_{0}{\bigl (J,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )} +\varepsilon H_{1}{\bigl (J,\theta,\varepsilon p_{i},\varepsilon q_{i};\varepsilon t\bigr )}\;. }$$ (12.1) Here, \((J,\theta )\) designates the “fast” action-angle variables for the unperturbed, solved problem \(H_{0}(\varepsilon = 0),\) and the (p i , q i ) represent the remaining “slow” canonical variables, which do not necessarily have…
Time-Independent Canonical Perturbation Theory
2001
First we consider the perturbation calculation only to first order, limiting ourselves to only one degree of freedom. Furthermore, the system is to be conservative, ∂ H∕∂ t = 0, and periodic in both the unperturbed and perturbed case. In addition to periodicity, we shall require the Hamilton–Jacobi equation to be separable for the unperturbed situation. The unperturbed problem H0(J0) which is described by the action-angle variables J0 and w0 will be assumed to be solved. Thus we have, for the unperturbed frequency: $$\displaystyle{ \nu _{0} = \frac{\partial H_{0}} {\partial J_{0}} }$$ (10.1) and $$\displaystyle{ w_{0} =\nu _{0}t +\beta _{0}\;. }$$ (10.2) Then the new Hamiltonian reads, up t…
Coordinates for quasi-Fuchsian punctured torus space
1998
We consider complex Fenchel-Nielsen coordinates on the quasi-Fuchsian space of punctured tori. These coordinates arise from a generalisation of Kra's plumbing construction and are related to earthquakes on Teichmueller space. They also allow us to interpolate between two coordinate systems on Teichmueller space, namely the classical Fuchsian space with Fenchel-Nielsen coordinates and the Maskit embedding. We also show how they relate to the pleating coordinates of Keen and Series.
Construction of canonical coordinates for exponential Lie groups
2009
Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Holder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ C Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g * . The cross-section mapping σ: Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras e 0 (Ω) and e 1 (Ω) of polarized and quantizable functions, respectively, a…