The motion of the classical and quntum partcles in the extended Lobachevsky

space

Yu. Kurochkin, V.S. Otchik, E. Ovseyuk, Dz. Shoukovy

Plan

Introduction

Classical problem

Quantum problem

Perspectives

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

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// ЯФ, 61, (1998) [6] V. Gritzev, Yu. Kurochkin// Phys. Rev B, 64, (2001)

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

2 2 2 20 1 2 3 02 2

2 2

{ , } { , }, , ,( ) ( )

1 1

r Rx x x x x x R r R

r rR R

2. Imaginary three dimensional Lobachevsky space outside of three dimensional sphere of three dimensional Euclidean space

2 2 2 20 1 2 3 02 2

2 2

{ , } { , }, , ,( ) ( )

1 1

r Rx x x x x x R r R

r rR R

Here are coordinates of points in the three - dimensional Euclidean space

{ }, 1,2,3jr r j

(1)

(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)

Spherical coordinates for the real

Lobachevsky space 1 1 1 0sin sin cos , sin sin sin , sin cos , cosh ,

tanh

u R h u R u R u R

x R

Spherical coordinates for the imaginary Lobachevsky space 1 1 1 0cosh sin cos , cosh sin sin , cosh cos , sinh ,

cot

u R u R u R u R

x R h

Metrical tensor of the real Lobachevsky space

2 2 2 2{1,sinh ,sinh sin }ikg R

Free particle (real Lobachevsky space)

1 2( , , , ) ( ) ( ),S t Et A S S 2

1 2( ) ,

sin

AS B d

Hamilton – Jacoby equation

Solution where

2 22

2 2 2 2

1 1 1

2 sinh sinh sin

S S S S

tmR

22 2( ) 2

sinh

BS mER d c

Free particle (imaginary Lobachevsky space)

Metrical tensor of the imaginary Lobachevsky space

Hamilton – Jacoby equation

Solution 1 2( , , , ) ' ' '( ) '( )S t E t A S S where

2

1 2

''( ) ' ,

sin

AS B d

2 22

2 2 2 2

1 1 1

2 cosh cosh sin

S S S S

tmR

22 2

''( ) 2 '

sinh

BS mE R d c

2 2 2 2{1, cosh , cosh sin }ikg R

Coulomb potential

1 2( , , , ) ( ) ( )S t Et A S S

2

1 2( ) ,

sin

AS B d

Real space. Hamilton – Jacoby equation

.

Solution

where

2 22

2 2 2 2

1 1 1coth

2 sinh sinh sin

S S S S

R tmR

22 2( ) 2 ( coth )

sinh

BS mR E d c

R

Coulomb potential

Imaginary space. Hamilton – Jacoby equation

1 2( , , , ) ' ' '( ) '( )S t E t A S S

2

1 2

''( ) ' ,

sin

AS B d

Solution

where

2 22

2 2 2 2

1 1 1tanh

2 cosh cosh sin

S S S S

R tmR

22 2

''( ) 2 ( ' tanh )

cosh

BS mR E d c

R

A charged particle in the constant homogeneous magnetic

field in the extended Lobachevsky space. Real space Metrical tensor is

Hamilton – Jacoby equation

.

Solution

where

22 2

2 2 2 2

1 1 1[ 2 cosh 2 ]

2 cosh cosh sinh

S S S SB r B

r z tmR z z r

1 2( , , , ) ( ) ( )S r z t Et A S r S z

21 2

2 cosh 2( ) ,

sinh

A B r BS r C dr

r

22 2( ) 2

cosh

CS z mER dz c

z

2 2 2 2{cosh ,cosh sinh ,1}ikg R z z r

A charged particle in the constant homogeneous magnetic

field in the extended Lobachevsky space. Imaginary space Metrical tensor is

Hamilton – Jacoby equation

.

Solution

where

1 2( , , , ) ( ) ( )S r z t Et A S r S z

2 2 2 2{sinh ,sinh sinh , 1}ikg R z z r

,

22 2

2 2 2 2

1 1 ' 1 ' ' '[ 2 cosh 2 ]

2 s inh sinh sinh

S S S SB r B

r z tmR z z r

21 2

' 2 cosh 2'( ) ' ,

sinh

A B r BS r C dr

r

22 2

''( ) 2 '

sinh

CS z mE R dz c

z

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= iρ

With Hamiltonian commute angular momentum operator

And analog Runge-Lenz operator , where

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

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

,

dR

nxnFdd

R

xnxnFdx

ii1

2

0

22

12

0

0

22

),(64

1),(

64

1)(

),( nin 12 n

nd Here

dxR

xnxxnF

i1

20)(),(

The inverse formula

where 0

3

2x

xddx - measure on the Lobachevsky space.

The analog plane wave 12

0

i

R

xnx

is the solution of the Schrodinger equation when

41

22 ER

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

,

In the imaginary space

dnxissbnFs

dR

xnxnFdx

s

i

)'()2exp()2;,(4

''),(

128

1)'(

12

12

0

0

22

The inverse formulas

'''

)'(),(1

20 dxR

xnxxnF

i

')'()2exp()'()2;,( dxnxisxsbnF

- is distinction from point 'x to isotropic direct line tnby

)'(cos bx

Quantum mechanical problem. Coulomb potential

Parabolic coordinates in the Lobachevsky space

1 20 1 1 2

1 2

2cos

2 (1 )(1 )

t tx x t t

t t

1 2 1 22 1 2 3

1 2

2sin

2 (1 )(1 )

t t t tx t t x

t t

1 20 1 0 0 2t t

Parabolic coordinates In the imaginary Lobachevsky space

1 20 1 1 2

1 2

2cos

2 (1 )( 1)

t tx R x R t t

t t

1 2 1 22 1 2 3

1 2

2sin

2 (1 )( 1)

t t t tx R t t x R

t t

1 20 1 1 0 2t t

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 1 1 2 2( ) ( ) imS t S t e 1S 2S

2 21

1 1 1 1 1 11 1 1

(1 ) (1 ) 02 4

dSd ER R mt t t t Sdt dt t

2 22

2 2 2 2 2 22 2 2

( 1) ( 1) 02 4

dSd ER R mt t t t S

dt dt t

where separation constants and obey the relation

1 21 2 R

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

2 2 21 1 1 2 1 1 1 1(1 ) ( 1 )m R RS t t F m t

2 2 22 2 2 2 1 2 2 2( 1) ( 1 )m R RS t t F m t

Here we have introduced the notations

2 2 21 1

2 2 22 2

2 21 2

1 2

1 12 2 4 1 2 2

2 2 21 1

2 2 4 1 2 22 2 2

1( 1 2 2 1 2 2 ) 1

2

mm ER R R ER

mm ER R R ER

R ER R ER m