Path-integral solutions for a class of 2D systems with local degeneracies

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Path-integral solutions for a class of 2D systems with local degeneracies

M. Bentaiba, M. Benkaddour, L. Chetouani, T. Hammann

To cite this version:

M. Bentaiba, M. Benkaddour, L. Chetouani, T. Hammann. Path-integral solutions for a class of 2D systems with local degeneracies. Journal de Physique I, EDP Sciences, 1994, 4 (1), pp.7-27.

�10.1051/jp1:1994118�. �jpa-00246891�

Classification Physics ^Abstracts

03.658 03.65D 03.65G

Path-integral solutions for

_a

Mass of 2D systems ^{with local} degeneracies

M. Bentaiba (l~

3),

M. Benkaddour

(1),

L. Chetouani

(2)

and T. F. Hammann (3.

*)

(~> Laboratoire de

Physique Théorique,

Institut de

Physique,

Université de Blida, Blida,

Algena

(2)

Département

de Physique

Théorique,

Institut de Physique, ^Université de Constantine,

Constantine, Algeria

(3) Laboratoire de Mathématiques,

Physique Mathématique

et

Informatique,

^Faculté des Sciences et Techniques, ^Université de Haute Alsace, 4 _rue des Frères Lumière, F-68093 Mulhouse Cedex,

France

(Received 30

April

1993, revised 19

Juty

1993. accepted 3

Septembei

1993)

Résumé. Le potentiel V(r) ar~~~~

pr~~~

_{est étudié par les}

_Intégrales

de Parcours. La fonction de Green _est calculée pour E

=

0 _et d

quelconque,

en coordonnées polaires et dans des coordonnées

généralisant

les coordonnées de Lévi-Civita. Il est montré que ^cette fonction de Green

est la _somme d'une

partie

discrète _mais finie _et d'une

partie

^{continue. Des} cas limites _et des _cas

particuliers sont étudiés.

Abstract. The potential V (r) ₌ ar~ ~~~

pr~~

~ is studied via the

path

integral approach. The Green function _is calculated for E

=

0 and for any value of d in polar coordinates _as well _{as in} coordinates

generahzing

the Levi-Civita transformation. It is shown that this Green function _is the

sum of _a discrete but finite part ^{and of} a continuous part.

Limiting

cases will be

investigated.

l. Introduction.

The aim of this paper is _{to use} the forrnalism of

path integrals

to calculate the Green function relative to the

potential

class defined

by

v(r)

=

r2d-2 prd-2, ₍₁₎

where _a and

p

_are

positive

parameters and d is _a

positive

or ml rational number. So, for d

= 1, V

(r)

= a p/r is the Coulomb

potential,

and for d

= 2, V

(r)

=

ar~ p

_is the harrnonic oscillator.

This two dimensional

potential

class has been

recently

studied

il -3],

and it _was

proved

^that the _state

corresponding

to E

=

0 featured _a

multiple

order

degeneracy

because of certain hidden symmetry properties.

(*> To whom requests ^for repnnts should be addressed.

It is shown that the Green function _can be calculated for any d and for E

=

0

by-first-using

the

polar

coordinates.

It is well known that, _say, for the Coulomb

potential,

the Green function _can be written _m compact ^{forrn via the} Levi-Civita transformation. Thus _we

generalize

this transformation for any d in order _to calculate the Green function. We _sum up the series obtained in

polar

coordinates

by showing

that it is

composed

of _two

parts, namely

_a discrete and finite _sum of Green functions _as well _{as a} continuons _one. We shall also consider _a few

limiting

_cases for

E=0 and deduce the energy spectrum ^{and the} wave functions for any E and for

d

= and 2.

2. Green functions _in

polar

coordinates.

In the canonical formulation of the

path integrals,

the propagator ^{is written}

forrnally

_as

follows,

in standard

notation,

K(ri

_r, ;

T)

=

j

s~x

s~p~

fl~y

s~p~

exp

l~ ^{j~ P~1}

^{+ P~,fi}

^~~

^~

^~~

^{V (x,}

y)j t~

*

(2)

o 2 _m

The radial

potential

class

(1) presents singularities

for ail values of the rational number de

[0, 2[.

Thus let _{us use} the Duru-Kleinert transformation

[4-6],

^which consists _m

choosing

an

adequate regulating

function

f(r)

_~ 0, _m order to avoid _a

path collapsing.

Let _{us use a t}

~ s time transformation defined

by

dt

= ds

f (r

=

ds

fi (r f~ (r

,

3

or m the discrete

version,

F = Es

fi (rn)f~(rn

i

),

T

=

(N

+ i

)

_s, s

=

(N

+ i ss,

where

fi

and

f~

are two

regulating

functions chosen

adequately

so as to stabilize the

path integral.

Using

the Green

function,

we now introduce the energy

E, setting

_m

=

h

=

for ail that follows,

m m

G

(ri

_{r~ ;}

E)

₌ dT K

(ri,

_{ri ;}

T) exp(iET)

₌ dS

P~ (ri,

_r~ S

,

(4)

o o

the _new form of the

P~

Kemel

being

~~

_~~~'

~~ _'

~~

2

«ÎÎÎ~ÎÎÎÎÎ~(ri) ^ÎÎi

₂

ÎÎÎ~f(r)

~~P~~~~

~

(5a)

where

N+1~(~

^{_~ )2}

Al

"

£

~

~ ~

~S

fi (rn

)

Iv (rn

E

f~ (rn

,

(5b)

n

~s

fi

(~n )

fr

(~n 1)

is the _new action.

An

appropdate family

of

regulating

functions _is given

by fÎ(r)

#

~f(r)j~

^~ and

fr(r)

#

~f(r)j~

Set A

=

0 _to

simplify

the

post-point prescription

calculation. The

stabilizing

functions _are

now

f~

m

f

and

f~

m

Having

_a central two-dimensional

potential,

it may seem convenient to go over to the usual

polar

^coordinate system, x = r cos o and y = r sm o ; yet, ^these coordinates _cannot prevent

path collapsing.

We therefore shift the

singularities

to

mfinity

with the transformation

(x, y)

_~

(q,

o

)

defined

by

the

following equations

_:

r=e~,

_-cc~q~+cc and 0wow2w.

(6)

With these _new

variables,

the action _can be _{wntten m terras} of the

followmg expressions

Ax~ ₌ x~ x~ _j = r~ cos

o~

r~ _i cos

o~

i _,

Ay~

₌ y~ y~_ _i = r~ sm

o~

r~ _i sin

o~

i ,

A~n ~ ~~~ ~~~

'

Aqn

_~ qn qn _-1

'

A°n

"

°n °n

-1

The

regulating

function

being

now defined as

f

=

~2q

the action is then

Al

=

~(~ ^[(Aq~)~

⁺

(Ao~)~]

_es

[a

e~~~~ p

e~~

_E

_e~~~]

+

A,~)

,

(7a)

~

2 ~s

where the correction

AA~

is

explicitly given by

i ~ 7

~ ~

(Aq~)~ (Ao~)~

_j

~

AAn

₌

j ^(Aqn)

⁺

ù ^(Aqn) (Aqn) (Ao~)

+

~

~ ^{(Ao~) (7b)}

Es

Let _{us now}

integrate

with respect to the

intervals,

instead of the x~ and y~

positions,

N

IN

⁺

fl dx~ dy~

₌

fl d(Ax~) d(Ay~)

n n 2

The Jacobian of this transformation

being equal

to

à

Ax~, Ay~

~~ ~ ~~

ô

(Aq~, Ao~)

^{~ '}

the _measure of

equation (5a)

is given

by

2

Ii

es

Î~ ~~Î~~~ÎÎ~~~ Î~

~

~ ~~

2

lies ^fi~ ~~Î~Î~ÎÎ~~~ fi~

~~ ~

~~~~~'

_~~~

wherem

C~~~

₌ ²

Aq~

+ 2

(Aq~)~

,

is the correction relative _to the _measure.

Taking

into _account the two corrections

odginating

in the action and in the _{measure, we now}

come to the total correction

C~,

(j~~

₎₂

C~

= i

AA~

^~ +

C~~~(1

+1

AA~). (9)

2

If _we miss out trie _terras

Aq~ ào~

^which do not contribute within the Es ~ 0 limit, _we obtain

(ARN

)~ ~

( _i(~~n)~

+

2(àqn)~ (~°n)~

₊

(~~n)~(~°n)~i

~ ~S ,

Cmeas àAN

_"

£ _12(Aqn)~

+

2(Aqn

_)~

(A%n)~l

By taking

^into account trie

relations,

((Aq)~)

= i Es,

( (ho )~)

₌ i es,

((Aq)~ (ho )~)

=

(i

_es

)~, ((Aq)~)

= 3

(1es

)~

i (A

o)4

)

=

3(1 _{es )2}

,

j (Aq

)6

)

= 15 (1 es )3

,

j (Aq

₎₂

(A

o)4

)

= 3 (1_es)3

,

j (àq

)4

(à

o)2 = 3 (1 ~s )3

,

correction

(9)

thus leads _{to an} effective nil

potential,

_V~~~

=

0.

Thus, from the above results _{we can}

readily

obtain

~ i ^{N + '}

d(Aq~) d(Aô~)

P~ (r~,

rj ;

(N

+

1) es)

₌

fl

x

2 "1ES

n =2

2 "1~S

and it follows that

l'E(rf,

_~i

~)

"

s 2

p2

=

Éqd~pù~ôd~p

g eXp dS pq ⁺ pg à ~

~

£t e~^~~ ₊

fl

e~~ ₊ E e~~

(Î1)

0

~ ~

Integrating

over the

angular

variable _{we are} led to

where the Hamiltonian H iS _a function of the

(q, s)

coordinates,

H=~ +«e~~~-fle~~-Ee~~ (12b)

Thus, _we have reduced the motion of the

particle Subjected

to the action of

potential (1),

to a

motion

governed by

the Sum of three

exponential potentials.

It iS obvious that the Green function _can be

analytically

calculated

only

for a combination of two

exponentials,

te- a

Morse hke

potential.

One _{can See} that there _are four _cases

admitting

an exact Solution

i)

First _case the

centnfugal

barrier, d

=

0.

In the

(q, s)

coordinate system, the

problem

of the bottier _is described

by

H=~

2

-Ee~~+ _{(« -p).} (13)

ii)

Second _{case :} the Coulomb

problem,

d

= 1.

In the

(q, s)

coordinate system, the Coulomb

problem

is descnbed

by

^the Hamiltonian related _to the Morse

potential [7],

iii)

Third _case the harmonic oscillator, d

= 2.

The Hamiltonian

describing

the H-O- in the

(q, s)

coordinate system is aise related _to the Morse Potential

[7],

iv)

Fourth _{case :} any d and E

= 0.

In this _case, the Hamiltonian

H=~ +«e~~~-fle~~, ₍₁₆₎

is aise related to the Morse

potential.

Let _us

analyse

this

important

4th _{case :} the Green function _can be wntten

~~~f,

_Î,

~) i ^gr

_~~P Il

Î(°f °i)] ~QfÎQI)

,

(Î7a)

f

~oe

with

iq~iqii

=

1"

^ds

j

5~q5Jp

exp

; ~

ds

jpq

^«

^e2àq

⁺

^{p eàq} j _(17b)

o o

With the

help

^of _{w =}

r~'~

= exp

[dq/2] transformation,

_{we see} that there _{is a}

simple

^relation

between

(q~[j)

and

(w~[w~)

_given

by [4], namely,

exp

(q~

+

qi)j ^iq~iqii

⁼

^{iw~ wii}

,

(18)

where

CC ~ ²

~~f ~i)

~

~~ ~~P

1~

J

^{fl ~~i~} w ~XP Il

~osc1

,

0 (~~)

with

A~~~ =

~

dS

P~

ùJ

~~

_{~ ~}

aw~

^~~

~~~~~

~~~

,

(20)

o

2 d 2

W

being

the

amplitude

related to the radial harmonic oscillator.

As _we know the

expression

of

(w~[w~) [4],

the Green function _can

finally

be

given by

~~~~ _~~

'

~

Ϋ

d"

o

~~ ~~~

~

~

d

~

id _sm

(2

S

(Id

^~

~ ~~~

~

~~~ ^~ ~~~ ^~~~ ~~ ~

~~~Î _~~

Î~ id~ill~~/d)

~~~~

We shall notice that, at this stage,

equation(21)

allows _us to deduce, for any

E, the Green functions relative to

i)

the Coulomb

potential

: we

just

have _{to set} d

= and

replace

_a with

(a E)

_: Gcoul

(~f,

ri

~)

"

i

~~~~~~ ^~~~ dS eXp

(4 1PS

^~

~~~

~

X

~oe

"

~

sm

(2

S

2(« E))

x exp [1 E

(r~

+ r~) cot

(2

S

E))] I~

_i

~

~

~~~~~~~~

(22)

sin

(2

S E

))

ii)

The two-dimensional harmonie oscillator _{: we}

just

have _{to set} d

= 2 and _to

replace

p with

fl

+E:

~°~~~~ ~~'~~

f

Î~ _2~"

^{~~~~~~ ~~~}

^o

^{~~ ~~~~~ ~~}

sin

ÎÙ)

~

~ ~~~

~

~~~ ~

~~Î

~~~ ~~

~~

_~

i

sÎÎÎ~)

_~~~~

Now it _is well known that there is _a compact ^{form for the} green function of the two-

dimensional Coulomb

potential [4].

This compact ^form can be obtained via _a Levi-Civita type transformation

generalized

for any d.

3. The Green function via the Levi-Civita type transformation.

3.1 GREEN _FUNCTION IN

(u,

u COORDINATES. Let _us consider the well known transforma-

tion Ii,

à)- (z, z*),

z=re'°

and

z*=re~'°,

as well _{as a}

(z, z*)

_-

(w, w*) transformation,

defined

by

w=~Éz~'~=u+iv, w*=/Àz*~'~=u-iu, ₍₂₄₎

The passage from

(x, y)

coordinates _to the _new

(u,

u coordinates _is given

by

^the

following

relations

~

d l'd (u _{+ iv} )2'd +

(u

_<& )2'd

]Id ~ ~ ~)21d

~

~ ~)21d

~

~~~

~ 2 21

Thus,

u =

~ r~'~ _cos

~,

~ _~

2

~4~2 ~~~ dé

~~~~

d 2

The _areas of variation for

(u,

_u

depend

_upon d and à. In matrix notation the system ^{of linear}

equations (25)

_may be written _as

r =

Bv,

_r

=

Ill

,

v

-

Ill

,

(27)

where

2 2 2 2

(u

+

iu)~

₊

_(u iu)~ (u

₊

iu)~ _(u iu)~

d ^na 2 2~

~

~2~

2

2~~

^{2 2}

u+iv)d

-(u-iu)d (u+iu)d +(u-iu)d

2i 2

For d

=

1,

the Levi-Civita transformation

~2 ~2

x =

,

y = uu, 2

can be

recognized.

With these _new

(u, u) coordinates,

the Kemel

P~(V~,

_V~;

S)

of the Green function

(4)

_can

now be

written,

P~ _{(V~, Vi}

_;

S)

₌

~~~~~~

~~~~~~

lim

jj

~f~ ^{d(Au~ ) d(Au~) J~}

_i x

2 _ari es gi

(V~) g~(V~

N _~ «

~ j

2 _ari es g(V~/

~ ~

~ ~~~

~ ~S ~i

~v/) _~r(vn

i

~'~ (un

+

iv~

~~ià ~~~ ~ ~ ~~~~

where

à

(àX~, ày~ )

_{d 21d 2}

~ ~ ~~

J~

_{i= =}

(u~

i+U~

1)

ô(Au~, Au~)

² ,

is the Jacobian of the

(x, y)

-

(u,

u transformation and g = gi g~ is a

product

of

stabilizing

functions.

Let _us eliminate the terni

g~(V~) gi(V~)[gi(V~) g~(V~)]~

^{within the}

mid-point prescription,

g "

gÎ

~

gÎ

~

J,1. (29)

Thus,

(u~

+

iu~)~ (u~

i +

iu~

_{1)~ =} ^{~ ~}

(û(

₊

à()~'d (Auj

+

Auj)

_x

2

(2 d)(2_3 d)

^{AU~ +1 AU~ 2} AU~ -1 AU~

j~

x +

~

+ +

(30)

~4 d

àn

+ 1Ùn

àn

l~n

and

Id

2

Au~

+

Au~ 2 Au~

i

Au~ j

~ ~

~ ~ ~n ~ l~n

~

~n l~n

~

~

~~~~

wherein

~ ~ ~ u + u

i

>i n-1

~ ~ ^{~ ~~}

Un ~

~ ^{~~ ~} 2

Thus,

_usmg

(28)

and

(29),

we obtain the _two

following quantities

in the exponent ^of

exponential

from

equation (28)

:

ii

the action

N + i

~ ~ d ^{21d 2}

~ ~ ~~~ ~

p

~ ~

~~

n~i

~ ~S

~~~~

~

~~~~

~ ~~

~

~~~~

_~ ~~~ ~

~ ^~ ~~~ ~ ~~~ ~~~~~

Let us notice that this action is

related,

for E

=

0,

to a two dimensional harrnonic oscillator whose motion _is

spatially

constrained.

ii)

a term groupmg

together

ail the fourth order terms m

Au~

^and

Au~,

~~

2

Îs ^{Î~~ ~~~~}

^~

^~~~~

^~

^~2 ~~ ~Î ^Î~~

^{~ ~} ~ ~

'

~~~~~

namely

the correction to the action

A~/.

At the limit Es - 0, it is easy to make _sure, thanks to

lAUjj

=

jàUl) (AUIA~>1)

~

l~~n~~nl

~ o'

that

(AA~/)

₌

0,

which _means that _no effective

potential

^is mduced

by

these

changes.

SO

finally,

the _new kemel is the

following

:

P~(V~,

_{V~ ;}

^S)

=

s~us~p~

s~us~p~ e~~~, (33a)

where

A~

₌

j~' ^S'(p~

^{ù + p~, ù ~~}

~ ~~

+ a

(u~

₊

u~)

₊ ^{~ ~} ₊

Ej ^(33b)

o

~

2

~

d

Let _us now

briefly

tackle two

particular

_cases :

i)

Ford d

= 1, ^the action is relative to the Coulomb

potential,

A)°~~ =

dS' p,~ ù _{+ p~ ù} ^{~~ ~ ~~}

(a E) (u~

+

u~)

₊ 2

fl

,

(34)

~'

^~

2

~

obtained via the well-known Levi-Civita transformation

~2

~2

~ 2 ^'

y=uv.

The Green function _is then

given by G~°~'(r~,

_r~;

E) oe

=

dS' e~

'~~'[ _{(V~ V~)}

₊

(-

_V~[

_V~)

,

(35a)

o

where

~~~~~~~

2 _{ar i}

sinll~l)

~

is the well known

[4] amplitude

relative to the harrnonic oscillator.

ii)

For d

=

2,

the action _is relative to the harrnonic oscillator

potential,

A(°

=

j~'ds'(p~

^{ù + pv ù}

^~~

^~

^~~

^a

^(u~

⁺

^u~)

⁺

^(E

⁺

^fl _)j

,

(36a)

o 2

obtained _via relations

x# u,

y= U.

The

corresponding

Green function takes the forrn

G~° _(r~, oe

r~ ; E

)

=

dS' e'^{~~ + ~l ~'}

(r~

_r~

,

(36b)

o

where

(r~[r~)

_is the usual propagator ^relative to the bi-dimensional harrnonic oscillator,

(Eq. (35b)).

In these _{two cases}

(d

= and 2

),

the Green functions _can be

easily

calculated. In the

general

case, for any d, the Green functions _can be

analytically

calculated for E

=

0

(Sect. 2).

When

going

from the

(x, y)

coordinate system over to the

(u, u)

_{system, we can see} that transformation

(25)

is not

univocal,

the _area of variation of _u and u is not the whole

plane

: it

depends

_{upon parameter} d and

angle

0.

It is thus not easy to find

G(r~,

_r~

0)

for any

d, according

to

equations (31, 33a, 33b).

TO

bypass

this

difficulty,

let _us go back _to

equation (21)

and _sum up the _serres to obtain

G(r~,

_r~

0)

_{as a} function of the _new

(u, u)

coordinates.

3.2 SUMMATION _{OF THE} GREEN FUNCTION for E

= 0. Let _us first notice that, here,

E is understood _as E +10.

According

to the

expression

of the action

(12c),

for E

= 0, the

imaginary terra 10 e~~

~ i0 _can be combined with ~

Î~/2

_to

_give

Î~/2

₁₀

-

(1

₊

10)~/2

-

Î~/2

₁₀

ii _[.

Let _{us now} take the

integral representation

of the modified Bessel function

[8], I~ (z

=

le~

^{~°' °} cos p 0 do ^~~~

~~

_"

e~ ^{~ ~°~ "~} dt

ar ,

j

ar

~

with arg

(z

_w ¹ and Re

(p

~ 0.

2

We _can

easily

prove that _{it can} be written

I~

(z ₌

£

_{dt e~ ~°~ + '~"}

,

(37)

"

Îc

where C represents ^the

following integration

contour

(Fig. I).

y

<c)

x

-n O *fl

Fig.

l. Integration contour for Bessel functions.

Let _us consider the sum

+ce ~oe

jj

e' ^~°

i~

_t

j~

(z

=

jj

^{dt e~ ~°~ ~~'~ '~ +' ~°}

,

(38)

t=-ce

f=-0e2"ÎC

2

ÀÎ _{rf~ rf'~}

v/he~e

z =

,

AH

=

0~ 0~.

id sin

(2

S

(Id)

One has

+ce +1 +ce

~

~i (Î ^{àfi +2t Î id)}

~

~iÎ(Afi ^{2nid) +10} _~ ~i Î(Afi t 2tld) -10

~ i

(~~)

i

,

f=-ce f=-ce f=i

where the _terras

±10,

_mtroduced beforehand, have been added in order to

regulanze

the

summations, and therefore

z exp[<(ÎAO

_+2t

[1[/d)- iii _0] ^=~

(cot

~~

-[+10)

_+cot ^~~

+É-10)).

ÎÎ~

2 2 d 2 d

Thus

~~~~

(~

~~~ ^~° ~2 Ii

"d~~~ 4~ar

~

~~ ~~ ^~~~

~°~ Î

~ ~~ _{~ ~°~}

Î

~ ~

Î'

^~~~~

Let _us define

J+

=

Î ldt

e~^~°~ _cot ^~~ ₊

L

± 10

(42)

4 " c 2 d

If _we

change

t _- t + 10 within

J~,

we shall obtain

J~

=

Î

_{dt e~}

~°~ cot

Î 1

,

(43a)

4 _"

ci

2 d

where

Ci

_is the

following

contour

(Fig. 2).

y

tc )

-fl*ÎÔ fl*ÎÙ

x

Fig.

2.

Integration

contour for Bessel function J~.

If _{now we}

change

t _- t +10 within J_, we shall obtain

J

=

Î

_{dt e~~°~}

cot

~~

É

,

(43b)

4 _ar

c2 2 d

where

C~

is the

following

_contour

(Fig. 3).

y

o ^x

+ .

-n-iO n-iO

(C2)

Fig. 3. Integration contour for Bessel function _J.

Eventually,

_we have

ÎΫ ^~~

^{~° ~~ 'd~~~ -}

h ^~

~ ~,

dt _coi

IA

^~

i

_ez Cos

=

1~

~ ~, ~~~~

where

Ci

U

C~

is the

following

_contour

(Fig. 4).

y

(ci> ci=Di u Ei u Fi

D' Fi

_~_~~+ ^{A El B}

~,j~*

x

~~ ~~~

E2 C n-iO*

D2 C2~D2 U E2 U F2

(c2) ~~

Fig.

4.

Integranon

contour for Bessel function Ic~ _~c~.

Given

F

(t)

=

e~ ^~°~ cot

~~

,

(45)

2 d

we can write

IF (t

) ^dt ₌ ^F

(t)

dt

=

Cl uC2

Dl+Ei+Fl+D2+E~+F~

1-

^{« <ce « + ire}

=

F

(t

dt ₊ F

(t

dt ₊ F

(t

dt

,

ABCD (46)

« + icc

«

icc

where

à ^~~~~

_{~ ~~~ ~~}

4~ar

~~

~°~

Î

_{~~~~~~ ~~~~~~~~}

The

potes

_t~ within the closed _{contour are} given

by

the

equation

AH t

~~~ ~

2 d ^'

their

positions

on the real axis

being

t~ ₌

(AH

2 _nar

d/2,

_n

integer.

The

potes

fulfill the condition

ar~t~~ar-- ar~

(AH -2nar)~~ar.

2

Thus

ÎÎÎ~

^{~~ ~~ ~°~}

Î

_{~~~~~ ~}

~^{~~~~ ~~}

,

From this formulation it becomes clear that

°~

~~~ ^~~ ~~

~~~~~

Î~

~~^{~~~ ~~} _~

~

~" Î- _Î+

^{cc ~~ ~°~}

Î

_{~~ ~~~}

+

+ dt cot

~~ e~~°~~

,

~~~Î (47)

2 d

Set _t

= ± _ar +

ip,

^then

11+

^{cc à} 7r 1p

z cash _p

jj~'~

^{~~ ~°~}

Î

_~~~~~~

-cc

~~

~~~

~ ^~ ~ ^~ ~ ^{~ ~}

This

gives

Î

~~~ ^~~ ~2₍₁

ld(Z )

"

j ₁

~~ ^{~~~ ~~} ₊

~

l~

^~

^dP

^{~ ~ ~°~~~}

_[C°t ~ _j

~)

i ~ ~ ~

-cot

(@+j-$)j. ₍₄₈₎

It _can

readily

be _seen that the Green function

(21)

can be

brought

into the form

~~~~

_~~'

~~

2

ar Î~

~ ~~~

~~

^~

~

id sin

ÎÎ~Î/d)

~

x exp

~ (rf

₊

rf)

_cot

(2

S

ÀÎ/d)

d x

2

/Î

cos t~

X

1~XP

~~~

~ ^(rfri)

⁺

n

id _SIn

(2

S 2

a Id

)

~

dÎ _Î-

ce

~~ ~~~

id sin

Î ~Î/d)

_~°~~

~ ~~~~~

~~Î

~ cos

(2

r/dÎ~~ ~~Î~Î

2

ip/d)

^~~~~

This set of

equations (48, 49)

becomes _more transparent ^if we introduce the new variables

#j, #~

^defined

by

t~=(A0-2nar)~=A#=#~-#~, #~=~0j

_and

#/=(0~-2nar)~,

(#/ depends

_upon

n)

and if _we note that _{we can}

always

find _{an axis} system ^{such that}

~~

do V

=

~ r~'~ ^~

=

U vn

=

~

~d12

C°S

#f u)

d do _u ^' d ^~

sjn~à

_n

~

Sin

Î

f u~

2

v~

₌

/@'2 ^{SIn ~Î~} _{#, ~tj}

=

(~'j

_v,

⁽⁵⁰⁾

Eventually,

the Green function _can be written _{as a} function of these _new

(u,

_u coordinates

~~~~' ~t'~~ à _Î~

_{~ ~'~~~~~~}

id

sinÎÎ~Î/d)

~ x

exp ~

((u))~

₊

(u))2

₊

^u2

₊

u2)

_cos

(2

S

ÀÎ/d) _Vi

_V

+

2 _sin

(2

S

(Id)

^{' '}

~

dÎ ^_

~

~~

~ ~~~~~~

cos

(2 arll~ ~~Î~Î

2

ip/d 1'

^~~~~

with

~ ~

~ ~ _~

(~

_{~ )d12}

~

~

f

d sin

(2

S

ÀÎ/d)

This

equation (51)

^is our main result.

The

G(r~,

_r~;

0)

Green function consists of _two parts : a finite _sum of Green functions relative to the bidimensional oscillators, and _a second part as

given by

the continuous

integral

let _us notice that this second part

disappears,

of _course, for d

= and d

=

2.

Let _us check

briefly

our result

(51)

_on two

particular

cases :

Case 1: d

= : Coulomb

potential.

In this _case, the number of t~

potes

^is two. Let _us repeat ^{here that} t~ =

~~

nar, and _as

2 0 _< AH _w 2 _ar, the condition is satisfied for _n

=

0 and 1. Thus,

o~ 0~ 0~

ibi=~, ibÎ~~, ib/"(Ùf-2")j"~-",

and

~ o

~

~ ~ ^i4Sfl

N~

~f, ~< _, "

~

~ X

n=o

0

17r SIn

(2

S

Î)

X ~XP

fl) (UÎ)~

+

(UÎ)~

+ UÎ ~

~Î)

_COS

(2

S

À~) _Vi ~<l,

(52)

2 sin

(

_a

or

ce

G

(r~,

_{r~ ;} o) ₌ dS

e~

^'~~

(V) _V~)

+

(VI V~)

,

o (53)

Wlth

V(

₌

Vi

Knowing G(r~,

_r~;

o),

it is

possible

to deduce

G(r~,

_{r~ ;}

E)

for any E.

We

only

have _to

change

_a into _a E. The obtained Green function will

correspond

to the Levi-Civita transformation.

Case 2 _: d

=

2 _: bidimensional harrnonic oscillator.

In this _case, the number of

potes

is _one. As _{ar w} AH

~ ar, the _ar

< t~ < ar condition will

only

be satisfied for _n

= o.

Hi 0~

Thus

#j=-, #~=-,

and

~~~°

_~~

'

~

o

~ ~~~~

2 1ar

SÎÀÎ)

~

x exp

~ ((u))2

₊

(u))2

₊

^u?

₊

^vi

_{) cos}

(S ÀÎ) _Vi .j

(54)

2 sin

(S ÀÎ)

The Green function for any E will be deduced

by changing fl

into

fl

+

E,

ce

G

(r~,

_{r~ ;} ^E

)

= dS e~~~~ +~~

(r~ r~)

,

o (55)

where

(r~[ ri)

is the usual propagator ^{of the} 2D harrnonic oscillator

(See

^Kleinert

[4]).

Let _{us now} go aven to the

study

of

limiting

_cases.

4.

Limiting

cases.

Let _{us set} E

=

o,

with E

=

E ₊

io,

and

taking (12c)

into _account let _us write

(17b)

in the

following

form _:

(Qf ~i)E

0,

f2

-

ii

_ds

_5Jq5~P

exP

i

lP4 -1

« e~~ ₊ P ^~~~ +

~°

₊

~°~ ~~~ 5

^~~~~

Under the _w

=

r~'~

=

e~~'~ transformation the

(q~ qj) amplitude

_can be obtained _as follows _:

(rf

_ri

)

E 0,

j~

~~ ~~~

~~

id Sin

ÎÎ~ÎÎd)

~

x exp

~ (r(

₊

r#)

_cet

(2

S

fild)) _i~

t jjà

~

~

~~~~~~~~

(57)

d id Sin

(2

S

(Id)

TO evaluate this

mtegral,

we use the standard formula

[9]

ldx

cet ^~ e~ ~ ~°~~ _~

J~ (

_a sinh _x

)

=

Î

2

~~

=

~~~]/~j~[~~j ~~w~~j~(Wfi+ p)M~~j~(WW-p),

where

W~,~j~(z)

and

M_~,~j~(z)

are the Whittaker functions. The formula is valid for

Re p _~ Re _a

(,

Re

(p/2

_v

~ l/2.

The

following

variable

changes

,fi

±

p

= ty~

~,

smh _x

=

(sinh y)~

,

cash _x

=

coth _y,

coth

(x/2)

=

e>,

coth _x

=

cash _y,

allow _{uS to} write

j~ siÎÎ

y

~~^~~ ~~~ ~~~ ~ ~~~ ~°~~

~Î ^~"

É~

=

~~

_~~

(j))~~-

^~

^wà,

_~i~

^{(ty~) Mi,}

_~i~

^(tyj

⁾

⁽⁵⁸⁾

where

W~ ~j~(ty~)

and

M~ ~j~(ty~)

are the Whittaker functions with y~ ~ y~ ~

0,

Ret~o,

[Argt[~ar

and Re

[(1+p)/2-A]~o.

~~~~~~~ ~

~ ~~

É~'

~

db'

^~

~

~~ ~~'

~

~

(r~

r~)~'~ ₌ t

/~,

p =

2

ii Id,

d

it follows that

~~f

~i)

E 0 ^"

r(1~ ^{(Î( P}

-2i 2 d

dÀÎ

~ ~

jj2/Î

àj

^{~ ~}

jj2/

d

À~ àj

(r~_r~)d'2 ^{r ~}

~

+ i ^d

à~'

~ ~ ^~~

d G' ~ ~ '~

'

(59)

5, Continuous states.

As _we noticed

before,

the

(10 e~~ Î~/2)

_term

can be wntten (-

(Î

₊₁

0)~/2),

which _means

that 10 is combined with

Î.

Now

fi

= ±

ii _[.

Let _us thus deterrnine the _wave function

by wnting

(o~-o~i

~ ^~Î

and let _{us use} the

followmg

relations

[10]

wà,

~

(z)

t

wà, ~(z), wà,

~

(z)

= ~

~(jB)~

~

Mà

~

(z)

+

~ ~~

~))

^~

~

Mà,

~

(zi.

Then _we have

~_~~~~ ~ ^~_~~~

~ °'

(2

_{ar )~}

/Î

~ÎÎ~ ^~~

~

r(

ⁱ

^{(Î( P} r(1~ ^{(f( P}

2 d

dfi

2 d

dfi

~

r(2 if (Id) r(1-

2

if (Id)

~dll~, ÎÎ

^~

@rd)

_W~ ~ ²

fi

~

dG'

^~

rf)

^~~~~

We thus obtain the _wave function for E

=

0

~~~~E

_-°

~

Ii

^~

^jf )[[)Ii ^{/1~ i}

_~~

~

x

~

_{W fl}

_W

^~

^~

r~) ⁽⁶¹⁾

r

dal'

à d

Or W fl

É1

^~

~ r~

_is

a linear combination

[10]

of

dal'

à d

~~'d~Î'l~~~~~~~

^~~~

_~~'d~l' Î~~~~~~Î

As _we have Il

M~, ~(z)

₌ ^{z" + ~'~} e~ ^~'~

iF

i

(p

A ₊

1/2,

2 _p + ; z

),

M~

~

(z)

= z~ ^{" + ~'~} e~ ^~'~

iF

i

(-

_p A ₊ 1/2, 2 _p + ; z

,

there _{are two states} for E

= 0 whatever d _:

#r~>(r)=e~~°r'~'e/~~~~~ifi(~j- '['+j,-~('+i;~firdj, ₍₆₃₎

where

ifi(a,

fl _;

z)

is the standard

hypergeometric

function.

These _states

~~+ l(r)

and

~~~ l(r)

are

linearly independent

unless 2

ii

Id _~ t4.

Taking

into _account the

following

forrnulae

[12]

in standard

notation, i~i(",

_Y

z)

" e~

ifi(Y

_{a, y}

z),

~~t~]~~~~"'~'~~~~~~(iiil _i~~~"

~~~

~'~~~'~~'

(which

implies

that d takes defined

values)

and

[13]

~~'n+@+l12,@(Z)

~

~

(~)~Z"~~~~e~~~~(2/l+1)(2/l+2).. (2/l+nilfl(~n,2/l+1(Z)=

~ (~ )~ Z" ^{~ ~~} e~ ^~~

LÎ

~ (Z

,

(64)

where

(n +1)

is _a natural number and

L("(z)

a

Laguerre polynomial,

it appears that in

equation (60)

~~~~~~~ dÎ~Î _~~'~~

~~°"

so that the constant in

equation (60) diverges,

~(rf) ~(ri)Î~~~~

_{Î "}

~(r) being

not norrnalisable, if will describe _a discrete bound _state.

The

single

~

(r)

wave function behaves hke

L(

^{"~ ~}

~

~

,

and in the _cases where

d d

=

and 2, it is calculable.

Part<cular _cases Case _: for d

= Coulomb

potential

~ _~~

= 2

iii

= integer d

In this _case

~

^~+ and ~ ^~~ are not

linearly independent

_{any more} and

equation (21)

takes the

following

form

+ce ii(fif~ fil) Gc~~i

(r~,

_r~

0)

=

z

^~

(t.~[r~)

,

(65a)

'= _-ce ^"

where

(~ j,, )

~~ ~4^{<ôS ~}

~

~

Î~

i sin

(2

S

+~)

~ ~~~

~~~~

~ G ~°t

(2

S

fi)

₁ ^{8 a}

^(r~

^{r )i12}

~ i _sin

(2

s

) (65b)

Let _us make _use of the standard formula

[9]

_in

equation (58)

and let _us set

y-21si, A-j, 21r=ty, 21(r~r~)i'2-ti, _~c-21ti,

we will obtain _:

~~~~~~ ~à ^{~+l~r(2[1[ ~~} ~à'~~~~~~~~~~à'~~~~~~~~

~

~

~ ~~~ ^~

(66)

The

potes

^of the gamma function _are

given by

Î~

~ ~ '~ ~~ ~

2(n

₊

Î[~+

l/2

)~'

where _a is the energy.

To _extract the discrete states, _{we use} the

following

formula

given by

^Kleinert

[5]

iim

(E E~) ru (E))

=

~~

,

E~E~

f'(E~)

ⁿⁱ

which is valid for

f(E~)

= n.

We thus obtain

~(r~) $r(r~)

=

~~

~ii(g~

^g~j ~ _~

n! grp

(r~ rj)i'2 r(2 if

₊ i

~à'

'~ ~~

~

~~~

~à'

_'~ ^~~

^{~ ~~~'}

^~~~~

If _{we use} the formula of reference

[4] again

~~'(1 _{~ M}Y2 ~ n, M12

(Zf)

_ÀÎ(1

+@)12 + n, M12(Zi

)

"

=

~)~~

^~ exp

(-

^{~~ ~~} (z~ + z~)~~ ⁺ ~Y~ M

(-

_n, + p,

zf) M(-

_n, + p, z~

,

(68)

and insert the

identity

(-

)n

r(-

_~

~

r(n

_{+ + ~}

r(-

_{n ~}

r(1

₊

~)

^' ~~~~

into

equation (68),

we

eventually

obtain the _wave function

~~~~ ~~~~~~~Î(2n+2[1[ÎÎ)~Î/+2[1[+1)Î~~~~

xeil°fille-àTr~jjij~2 jr), ₍₇₀₎

where the

Lf(z)

functions _are the usual

Laguerre polynomials [14]

Lr(z)m

^~~

) ()'M(-n,

~c

+1;z). (71)

Case 2 _: for d

=

2 _: harmonie oscillator

potential.

In this _case, equation

(21)

becomes

GHO~~f,

_~<

'

~~ ~~ '~~~7r

^~~~

~~~~

~'~

~

~~~~~

where

ce

/~

~~~~~~

°

~~ ~~~~

i sin

(ÎÙ)

~

X ~XP (1

À~(rÎ

~

~Î)

_C°t

(S À~))

_Î

Î

~j~

(72b)

SIn a )

JOURNAL DE PHYSIQUE i T 4, N' i IAN~ARY 1994 2