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arXiv:hep-th/9802098

Example of quantum systems reduction

J. Manjavidze

Published Feb 13, 1998• 0 citations

Abstract

To solve the quantum-mechanical problem the procedure of mapping onto linear space $W$ of generators of the (sub)group violated by given classical trajectory is formulated. The formalism is illustrated by the plane H-atom model. The problem is solved noting conservation of the Runge-Lentz vector $n$ and reducing the 4-dimensional incident phase space $T$ to the 3-dimensional linear subspace $W=T^* V\times R^1$, where $T^* V$ is the (angular momentum ($l$) - angle ($\vp$)) phase space and $R^1 =n$. It is shown explicitly that (i) the motion in $R^1$ is pure classical as the consequence of the reduction, (ii) motion in the $\vp$ direction is classical since the Kepler orbits are closed independently from initial conditions and (iii) motion in the $l$ direction is classical since all corresponding quantum corrections are defined on the bifurcation line ($l=\infty$) of the problem. In our terms the H-atom problem is exactly quasiclassical and is completely integrable by this reasons.

DOI: 10.48550/arxiv.hep-th/9802098

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To solve the quantum-mechanical problem the procedure of mapping onto linear space

W

of generators of the (sub)group violated by given classical trajectory is formulated. The formalism is illustrated by the plane H-atom model. The problem is solved noting conservation of the Runge-Lentz vector

n

and reducing the 4-dimensional incident phase space

T

to the 3-dimensional linear subspace

W=T^* V\times R^1

, where

T^* V

is the (angular momentum (

l

) - angle (

\vp

)) phase space and

R^1 =n

. It is shown explicitly that (i) the motion in

R^1

is pure classical as the consequence of the reduction, (ii) motion in the

\vp

direction is classical since the Kepler orbits are closed independently from initial conditions and (iii) motion in the

l

direction is classical since all corresponding quantum corrections are defined on the bifurcation line (

l=\infty

) of the problem. In our terms the H-atom problem is exactly quasiclassical and is completely integrable by this reasons.

We consider two-dimensional harmonic oscillator in the complex Bargmann-Fock-Segal representation with

T^*{\mathbb R}^{2}={\mathbb C}^2

as classical phase space. We show that the eigenfunctions

ψ_n

of the quantum Hamiltonian correspond to complex radial coordinates in the reduced phase space

{\mathbb C}^2/{\mathbb Z}_n\subset{\mathbb C}^2

. They describe

{\mathbb Z}_n

-invariant motion of particle along a circle

S^1

in lens space

S^3/{\mathbb Z}_n\subset{\mathbb C}^2/{\mathbb Z}_n

, where

{\mathbb Z}_n

is the cyclic group of rotation by an angle

2π/n

on the circle

S^1

,

n=1,2,...\,

. Thus the general solution of the Schrödinger equation carries information about an infinite number of admissible classical states

ψ_n

that can be mapped to other states after lifting into the quantum bundle. We show that in the Kepler/hydrogen atom problem there is a similar correspondence between classical and quantum states.

Paper contains description of the fields nonlinear modes successive quantization scheme. It is shown that the path integrals for absorption part of amplitudes are defined on the Dirac (

\\d

-like) functional measure. This permits arbitrary transformation of the functional integral variables. New form of the perturbation theory achieved by mapping the quantum dynamics in the space

W_G

of the ({\\it action, angle})-type collective variables. It is shown that the transformed perturbation theory contributions are accumulated exactly on the boundary

\\pa W_G

. Abilities of the developed formalism are illustrated by the Coulomb problem. This model is solved in the

W_C

=({\\it angle, angular momentum, Runge-Lentz vector}) space and the reason of its exact integrability is `emptiness' of

\\pa W_C

.