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1
The motion of the classical and quntum partcles
in the extended Lobachevsky space
Yu. Kurochkin, V.S. Otchik, E. Ovseyuk, Dz.
Shoukovy
2
Plan
Introduction
Classical problem
Quantum problem
Perspectives


3
Introduction

• Quantum-mechanical problems in the spaces of a
  constant positive and negative curvature are the
  object of interest of researchers since 1940,
  when Schrödinger was first solved the
  quantum-mechanical problem about the atom on the
  three-dimensional sphere S3. The analogous
  problem in the three-dimensional Lobachevsky
  space 1S3 was first solved by Infeld and Shild
  and imaginary Lobachevsky space C. Grosche
  (1994). These authors found the energy spectrum
  to be degenerate similarly to that in flat space.
• In recent years the quantum-mechanical models
  based on the geometry of spaces of constant
  curvature have attracted considerable attention
  due to their interesting mathematical features as
  well as the possibility of applications to
  physical problems

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Introduction

• An additional constant of motion, analog of the
  Runge-Lenz vector for the problem on the sphere
  S3 and for Lobachevsky space 1S3 together with
  angular momentum generate algebraic structure
  which may be considered as a nonlinear extension
  of Lie algebra, and which was called cubic
  algebra 1,2,3,4,5.
• Kepler-Coulomb problem on the sphere S3 has been
  used as a model for description of quarkonium
  spectrum, and ecxitons semiconductor quantum
  dots 6 .
• 1 P. Higgs// J. Phys A. Math. Gen., 12, 309,
  (1979)
• 2 H. Leemon J. Phys A. Math. Gen., 12 , 489,
  (1979)
• 3 Yu. Kurochkin, V. Otchik// Dokl. Akad. Nauk
  BSSR, 23, (1979)
• 4 A. Bogush, Yu. Kurochkin, V. Otchik// Dokl.
  Akad. Nauk BSSR, 24, (1980)
• 5 A. Bogush, Yu. Kurochkin, V. Otchik// ??,
  61, (1998)
• 6 V. Gritzev, Yu. Kurochkin// Phys. Rev B,
  64, (2001)

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The interpretation of the three dimensional
extended Lobachevsky space in terms of three
dimensional Euclidean space
As is well known there exist interpretations (F.
Klein, E. Beltrami) of the three dimensional
spaces of constant curvature in terms of three
dimensional Euclidean spaces. These
interpretations provide in particular
applications of the quantum mechanical models
based on the geometry of the spaces of constant
curvature to the solution of some problems in the
flat space. For example the following
interpretation of the three dimensional
Lobachevsky space can be used
1. Real three dimensional Lobachevsky space
inside of three dimensional sphere of three
dimensional Euclidean space
(1)
2. Imaginary three dimensional Lobachevsky space
outside of three dimensional sphere of three
dimensional Euclidean space
Here are coordinates of points in
the three - dimensional Euclidean space
(2)
R - radius of sphere in the Euclidean space and
radius of curvature in the Lobachevsky real and
imaginary spaces in the realization defined by
formulas (1),(2)
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Spherical coordinates for the real Lobachevsky
space
Spherical coordinates for the imaginary
Lobachevsky space
Metrical tensor of the real Lobachevsky space
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Free particle (real Lobachevsky space)
Hamilton Jacoby equation
where
Solution
8
Free particle (imaginary Lobachevsky space)
Metrical tensor of the imaginary Lobachevsky
space
Hamilton Jacoby equation
where
Solution
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Coulomb potential
Real space. Hamilton Jacoby equation
Solution
where
.
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Coulomb potential
Imaginary space. Hamilton Jacoby equation
Solution
where
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A charged particle in the constant homogeneous
magnetic field in the extended Lobachevsky space.
Real space
Metrical tensor is Hamilton Jacoby equation
Solution
where
.
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A charged particle in the constant homogeneous
magnetic field in the extended Lobachevsky space.
Imaginary space
Metrical tensor is Hamilton Jacoby equation
Solution
,
where
.
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QUANTU MECHANICAL PROBLRM

• The Schrödinger equation for Kepler-Coulomb
  problem on the sphere S3 and in the Lobachevsky
  space 1S3 is
• 
  where
• xµ are coordinates in four-dimensional flat
  space.
• R is a radius of the curvature for 1S3 R i?
• With Hamiltonian commute angular momentum
  operator
• And analog Runge-Lenz operator
• 
  
• 
  , where

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QUANTU MECHANICAL PROBLEM

• Operators Ai and Li obey the following
  commutation relation
• The energy spectra of the Hamiltonians are
• ? S3
  space
• n is
  the principal quantum number
• 1S3 space ?

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Gelfand-Graev transformation of the wave function
in the real Lobachevsky space



Here
The inverse formula
where
- measure on the Lobachevsky space.

The analog plane wave
,
is the solution of the Schrodinger equation when
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Gelfand-Graev transformation of the wave function
in the imaginary Lobachevsky space


In the imaginary space

The inverse formulas

to isotropic direct line
- is distinction from point
,
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Quantum mechanical problem. Coulomb potential

• Parabolic coordinates in the Lobachevsky space

Parabolic coordinates In the imaginary
Lobachevsky space

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Solutions of the Schroedinger equation in
imaginary Lobachevsky space

• Substitution
  separates the variables and equations for
  and
• in the case of imaginary Lobachevsky space are

where separation constants and obey the
relation

Solutions of these equations can be expressed in
terms of hypergeometric functions


Here we have introduced the notations