Do Atomic Spectra Determine Energy Levels?

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Do Atomic Spectra Determine Energy Levels?

C. Billionnet

To cite this version:

C. Billionnet. Do Atomic Spectra Determine Energy Levels?. Journal de Physique I, EDP Sciences,

1995, 5 (8), pp.949-961. �10.1051/jp1:1995175�. �jpa-00247119�

Classification Physics Abstracts

03.65 32.00

Do Atonùc Spectra Deternùne Energy Levels?

C. Billionnet

Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau Cedex, France

(Received

7 December 1994, received in final form 13 April 1995, accepted 18 April

1995)

Abstract. We examine the question of attributing _energies to atomic excited states, given

experimentalline

shapes. It is underlined that potes ofthe resolvant operator _are not determined by peaks _in cross~sections. Moreover, _in the scattering of

one photon by _an atom, resonance

energies seem to depend _on final products.

1. Introduction

When _one deals with the atom in

ordinary

quantum

mechanics,

for

example using

^{the Coulomb}

potential,

the levels _are in _a one-to-one

correspondence

with the

poles

of _the resolvant of the

Hamiltonian. This

correspondence

is

usually

conserved when the

coupling

with the quantum

electromagnetic

field is introduced: _one then expects ^the

poles

to be

displaced

into the

complex plane.

For such excited states, ^the usual mathematical

description by

state vectors is _no

longer possible (see

however Ref. _[Si

).

We _are

only

left with these

poles (one

must note here that the

specification

of 2

poles gives

less information than _a

spectral line).

So the

description

of the level is modified in the

following

_way: the real and

imaginary

parts ^of the

pole correspond

respectively

to the energy and width of the state; ^the _energy ^is shifted from the "nov

perturbed"

one due to radiative

corrections,

and the width accounts for the

instability.

But

experimeut gives

spectra, not

poles;

in fact

energies

_are associated to atomic levels

by

_meaus of

peaks

in spectra. So _{we want to} examine here whether this

procedure

of

deducing

excited states from spectra,

together

with their

characteristics,

has _a theoretical foundation.

We _use the

scatteriug

of _a

photon by

_an atom in its fundamental state. First _we consider i

photon

final _states. We _suppose that the

incoming photon

_{energy is} close to _a resonant

value, corresponding

to a

giveu

^excited state. We show that the real part of the resolvant's

pote

is

likely

to be different from the _sum of the _energy

(with

radiative

corrections)

of the fuudamental state of the atom and that of the maximum of the

peak. Thus,

that real part caunot be

directly

measured.

Nevertheless,

_{cau one} define the excited level _{energy as} the

preceding

^sum? The

answer seems to be _no when _one looks at the

peak given by

another scattering

amplitude,

the

one

involving

2

photons

in the final state, with the _sanie

incoming photon

_as

above;

the atom is in the fundamental _{state in} both _states. This

amplitude

is not considered

generally,

because

one Iooks

only

at the first orders of

perturbation theory; nevertheless,

it must be looked _at if

© Les Editions de Physique 1995

exact values of transition

probabilities

_are looked

for,

and that is _{our concern}

here;

_vie want to go

beyond

the electric

dipole approximation.

We shall _see that the

peaks

of the two

probability

densities do _{not seem} to be located at the _{same energy.} That will lead _us to the conclusion that there is _a

dilliculty

_m

representing

an atomic level

by

_a

quantity

extracted from

experiment.

Dur

plan

is _as follows: in Section 2, _we introduce the

Hamiltonian,

the

asymptotic

states and the

probability

densities for the

scatterings

we consider. In Section 3, the

problem

of the

relation betwween _a peak and _a

pole

is introduced

first,

in the

example

of the production of1

photon (Section 3.1).

Then _we show

(Section 3.2)

which approximations lead to the

equality

between the abscissa of the

peak's

maximum and the real part ^{of the}

ampIitude's pote.

In Section 3.3, _we show how this

equality

is

probably

invalidated when _one takes account of corrections to these

approximations.

In Section 4, we show that there _{is no reason}

why

the

peak

in the

probability density

for the

production

of 2

photons

should be located at the same

value _as the _one for the

production

of

photon

2. Trie

Hamiltonian,

trie

Asymptotic

States and trie

Probability

Densities

2.1. THE

INTERACTION,

ITS HAMILTONIAN AND THE FINAL STATES. The atom-field interaction is

described,

in the Coulomb _gauge,

by

the Hamiltonian H:

H=Ho+Hint

Hint " HI + H2

Hi =

~ ^~c

^~

_P~.A(r')

>

~~ ~ ^ÎÎÎ~2

~~~~~'~~~~~

Air)

=

Ai(r)

=

/

dk

~j

Aqkj(Àk,OEak,>~,«

e'~'~

₊

Àk,aa(,

_>~

~e~~~'~)]

~

Aw(ki =

(~j(~j)~~~

One _{uses, as} asymptotic states, wave

packets

constructed with states

if)>~

^{introduced in}

reference [3].

They

_are,

formally, eigenvectors

of H with

eigenvalues Ei Î hwk (Ei

_is the smallest

eigenvalue

of

H). They

_are

supposed

to descnbe "a

photon

with _wave vector k and

polarization

Àk

accompanying

the atom in its fundamental state". In the _{same way,} 2

photon

states _are described

by superpositions

of states

WI

> ,~ > [3].

2> k2> 3> k3

2.2. TRANSITION AMPLITUDES. Let _us first consider the

scattering ~/+A

_-

~/+A,

the atom

being

m its fundamental state before and after the interaction. So _we have two

asymptotic

states

containing photons ki,

Àki

(incoming)

and k2,_Àk~

(outgoing), respectively.

The _cross~

section is known if _one knows the function Tki,k~,>~~,>~~, ^defined by

~ ~k2,À~~

~Îi,À~~

^~~ ~(ki,Àkiik2,À~2)

~~~~(Î~IÎ

Îk21)l~ki,k2,Àki,À~~

We denote

by (

=

à[kilc

=

à[k21c

^the common energy of the 2

photons.

One obtains the

following expression

for T2-1

lin

abreviated

notation)

[2, 3]:

T2-1

= <

titi

_jHin~,

a2iiEj

+

( H)~~jHin~,ajj titi

>

+ <

Ii

lH>nt,

an iEi Î H)~~lH>nt,

_a21

Ii

>

il)

+ <

Ii

liH>nt, a21,

ail Ii

>

This function will also be denoted

by Ti,

the

subscript

1

indicating

that there is I

photon

m the final state; _a~ is the annihilator of _a

photon

with _wave vector k~ and

polarization

_Àk;

Commutators in this formula _are

given by:

ii~~~, _a~j «ii

=

i _ill~ AwiAw~ÀkiÀk~

^e~~~~~~~~'

~

[Hi, ail

=

£

^~

~

àfi

A~~e~kirÀ~~ vi

Î

~

Not

only

does

T2-1 depend

_on

(,

but also _on

ki,

_k2> Àki> Àk~. ^Let us fix the directions _ni and _n2 of ki and

k2,

_as well _as Àki ^and _Àk~ ^the

only

variable is then

(,

and it is that function

off

that _we denote from _{now on}

by

T2-1. The diiferential cross~section

da2-1

in _a solid

angle

element is

given by:

j~~)4

d0E2-1(1)

"

~Î1~2-1(1)Î~ÎÎÎ~~Î

We denote

by

_pi

(()

the

probability density

for this I

photon

transition.

At the lowest order of the

perturbative development

with respect to Hint, the 3 terms of

il) correspond respectively

to the

following diagrams:

,,,, diag. ,,~' "'É<~,,,,, ^~a8. ],,,~'(" _lj~,, ^~°"

^~

,,

"~2

k,

',,~~

k2

jjj,,fl~ _~,,"

r

""'

e

""

f r

""'

©

"""

r ~

"' "'

~

Fig. 1.

As for _{~i +} A _- 2~i +

A,

A still

being

_in the fundamental state, ^the cross~section is known from

T2,3-1,

which will be considered in Section

4;

it is defined

by:

~ ~k2,À~~;k3,À~~

flÎÎi,&~

^{~ " ~~~l~ ~(À~°2 + àW3}

Î)

_1~2,3-1

We shall still _suppose the direction and

polarization

^{of the}

incoming photon

_as

gjven. T2,3-1

is _a function

off

and (Àk~, Àk~). One has

T2,3-1

_{" < ÎÉf}

Î~int,a2a3j flÎÎi,&~

^~

The diiferential cross~section is

given

_[3]

by:

j~~)4

~0E23,1 "

jÎT2,3-lÎ~ à(àW2

+ àW3

Î) ~~2 dk3

We shall

give

_{a more}

explicit expression

^of

T2,3-1

in Section 4.I., as well _as the

diagrams

contributing

to the lowest order.

3. Relation Between _a Peak and _a

Pole,

in 1-Photon Production

3.1. THE PROBLEM. CAN ONE DEDUCE _THE REAL PART OF A RESOLVANT'S POLE FROM

A RESONANCE PEAK?. We want to examine the

peaks,

_in the

( variable,

of the

probability density

_pi

If

for

getting

_one

photon emitted;

with

n(()

_a

normalisation,

_pi

If

=

n(() [T2-1If)

Î~

The

problem

is to compare the

poles

of the resolvant G with the

position

of the

peaks.

It would be

universally acknowledged that,

in the

neighbourhood

of the _non

perturbed

_energy

Ee

of _an excited _{state e,} G has _a

pole

in _some sheet of the

complex plane;

let _us denote its value

by E(.

If

(

₌

(

+

in

denotes the

complex photon

_energy, the

pole

of

G, by il ),

leads to the function

(

-

Ti(() having

_a

pole

for

(

₌

E( Ei

_:=

(e.

In turn, it _{causes a}

peak

of

(

_{- pi}

If)

for

a value

((

~

of

(, diifering

_a

priori

from the real part ^of

(e,

denoted

by je.

Let _us suppose that

Ti(()~can

be written _as

(( (e)~~q~i,e((),

with q~i,e

regular (simple pole).

The

"position"

fi

~

of the

peak

of _pi in the the

neighbourhood

of

je depends

of _{course on} that function q~i,e.

Ai

a consequence, the

precise position

of the

peaks

of spectra ^does not

only depend

_on the resolvant's

poles.

This _was noted in reference [4]. ^{To illustrate the} truth of

this,

let _us note that if

n(()[q~i,e(()Î~

₌ I, thus

pi(()

=

III (e)~

+

q()~~

and pi has _a maximum for

(

₌

je.

This is the usual form of the

probability density,

obtained

by

the approximate calculus recalled

in Section 3.2. But ifn[q~i,e_Î~ ^{diifers from} 1, the

peak

_may be shifted at _a value

fi

~

# je,

_as it is the _case if çoi,e

If)

₌

(

_{ou çoi,e}

If)

_"

(~.

We _are thus driven _to try ^{and determine}

Îhe

_function

§~1,e

3.1.1. Trie Need for

Approximations.

An _exact determination of that function çoi,e ^{is in} fact diilicult for _{many reasons.}

First,

formula

il

which

gives

Ti contains 3 terms; next one has to use a

perturbative expansion

of the resolvant and to introduce _every excited state of the atom as intermediate states. This leads to an

infinity

of

diagrams. Lastly, given

an excited state, the

corresponding pole

is not known

explicitly; E(

_can

only

be calculated in _an

approximate

way

(The

function which has

E(

_{as a zero is}

given

in

Paragraph 3.3.1.).

As _a consequence of these points, _one is led _to consider

approximations

for Ti and

(e.

We shall _use, in the

following,

3 approximations which, for

simplicity,

_we shall denote

by

Appo>

Appi>

and

App2. App2

is the

finest, Appo

the

roughest

and

Appi

intermediate. We shall _see in Section

3.2., that,

^when

approximation Appo

is

used,

the abscissa of the maximum of the

(approximate) probability density

coincides with the real part ^{of the}

(approximate)

amplitude's pole.

But there is _{no reason}

why

this property ^{should still be} true with the

approximate amplitudes

in

Appi

_or

App2,

which _are doser to the exact _one. That is the

content of

conjecture

1. It must be

recognized

that the calculation _on which the argument

is based is not

precise enough

to exclude

completely

_a

position

^of the

peak exactly

at

je;

(besides

the diiliculties

just mentioned,

_an other _one is the exact

computation

of the _zero of the derivative which

gives

the

maximum). However,

should it be the _{case, a}

proof

would be

essential.

3.1.2. Trie

Approximations. Approximation App2

cousists in

replacing

Ti

by

_some of the resonant terms, ^{in the}

neighbourhood

of the considered _energy;

they

_are obtained

by expanding

G and fil in powers of Hi _up to first orders

(after

_some

resummations). Appt

is obtained from

App2 by keeping only

^{the lowest order} term.

Appo

is obtained from

Appi by replacing

the

pole (e by

an

approached value,

denoted

by le.

_{Moreover, we} make the

large wavelength

approximation.

3.2. THE APPROXIMATION

App0,

IN IÀÎHICH _THE PEAK _IS LOCALIZED _{AT THE} REAL PART

OF THE POLE OF Tl

3.2.1. Trie Resouant Terms in trie

Neighbourhood

of _a Peak. The _euergy of the

iucoming photon

is

supposed

to be closed _to trie _resonance

corresponding

to a

given

excited state _e.

In ail trie

approximations

_we

consider, only

the _resonant terms _are taken into _account. More

precisely,

_we shall

forget

trie last 2 terms in equation

il),

denoted _as follows:

~~~~

j ~~-i

_iH>n~,

_a~i

_{fl~i >}

Tzci

_#

j~[j _il~l)~i ^ii>~~, _{a~i, ail lli}

>

,

though they

contribute _{to çoe, even} if

they

_are not

singular

at

(e.

We denote

by TÎ-1

" ~ fIÎÎ

ÎHint,a21(~/

+

Î H) ~(Hint>a~Î

fIÎÎ ~

trie term _we

keep.

This

simplification

is

justified by

the fact that this term dominates at the lowest order.

As _{we are}

looking

for trie behaviour _neon Ee

Ei

(Ee ^{is the real} part of

E(,

close to

Ee)

,

it is natural to introduce the

special

vectors of the Hilbert _space that the eigenvectors _e,

e',

of the atomic Hamiltonian _are. For the chosen state _{e, we} set

T)=-<fil[[Hint,a211e> <e[(E/+(-H)~~[e> <e[[Hint,a(][fil>

One has

T[_i

₌

T)

+

T(~,

^where

'

Té~ "

L

_<

_{if lHint,~21}

_{11' >} <1'

(El

+

( H)~~

_{l1" >}

<1" lHint,~ll if

~

~i~m

1' and

1",

which _{may or may not} contain

photons,

_are not both

equal

to _e. In ail the

following approximations,

we

keep only T).

3.2.2.

ApproYimation Appo.

In order to obtain Appo> _we start

by replacing

Ti

by T),

which does _not

modify

the

pole (e

but _may shift the abscissa of the

peak's

maximum; _we denote

by

_([~~ this abscissa

(varying

with the next

approximations).

We also

replace

fil

by f. Finalli,

_as neither

([[~

nor

(e

_are

easily computable,

_we

replace,

in the

expression

(El

+

( Ee Re (El

+

())~~

for _{< e}

(El

+

( H)~~

_e > [2], ^{the function}

Re(E/

+

() by Re(Ee).

This leads, for the

pole,

to the

approached

value

((e)~pp.

One thu8 gets the

approximation

Appo

for Tii _we denote it

by Tΰ~

~Î~~ _" ~

Î Î~int>a21

e >

~( ((e)app)

^~ < e

Îflint>a~Î Î

~

That function

corresponds

to

Diagram

1in

Figure

1, except

that,

in the propagator, ^the non

perturbed

_energy of _e is

replaced by

the

complex

_energy

((e)app.

We deuote

by

pΰ~ the

corresponding density. Setting Re(Ee)

₌

àAe -1)re,

_one has:

((e)app

₌ Ee ₊ àAe

Ei 1~re

and

~~~~ ~~~~ ^~

~

~Î ^~ÎÎe ~ÎÎÎÎ~ ~ÎÎ)~ÎÎÎÎ ^~

^{~ ~}

Making

_now the

large wavelength approximation,

_we

replace

e~~~

by

1 in the values of the commutators.

n(() being proportional

to [([~, we get _a

peak

centered at

Îl,e

l~e +

ÎÎàe ~f (Îe)app

So there is _an

equality

between the abscissa of the

peak's

maximum and the real part of the

pole

of the

amplitude approximating Ti

in

Appo.

Now let _{us see} what diiference it makes _to take into _{account some} of the

previously neglected

terms.

3.3. DISCREPANCY BETWEEN _THE LOCATION _{OF THE} PEAK _{AND THE} REAL PART _OF THE

POLE _OF

Tl

IN APPROXIMATIONS

Appl

AND

App2

3.3.1.

Discrepancy

in trie

Appro~imation Appi Appi

_is defined

by conserving

the function

Re(z);

_we do not

replace

it

by Re(Ee).

Tl~~il)"~<fiiHint>~211e> <eiiEj+Î~Ee~ReiEt+Î))~~ie>

_X

< e iHint>

ail f

>

Then,

not

only

_is the

pole

shifted at _a value

(e

diiferent from Ee + àAe

Ei ià),

but

(Ei

+

( Re(Ei

₊

())~~

_cannot be written any more as

(( (e

)~~ So the

position

of the

peak

may ^not coincide with the real part ^{of the}

pole.

3.3.2.

Discrepancy

in trie

Approximation App2.

Let _us define _now

App2,

^for Tj we shall denote it

by TÎ~I.

To

TÎ~I,

defined

above,

_we add corrections

T)§1,

T)§1 ^and

T)§1, given by

^the

following

to 3

diagrams,

of fourth order _in

Hi1

_one should

allo adÀ

_the

syJimetrical

_ones:

~,l"

~ ~ ,,,"

"'l[ _tliag.

_l

_,,L,,

,,,

',,,ki _diag.

2

,,,"'

",,, _,,' ,,",

',,,

,,,'[~"'j,,,

",,i .,' Î ", ,"'

." ',

r ° ~2 ~l f f _e ez ej f

diag.

3

~

diàg.

4

"',

_~l

~

k2,"' "",1

,,,~-"-~,,,

~l""

,' k _'

',

," ^' " "

', ' ',

" '/, '"'

', ( _'

~" _" "', "' ",

f ~i e j ^e f f e~ _~ e~ ^f

Fig. 2.

We show in the

appendix

how to obtain the first of these

diagrams

in _a

perturbative

expansion.

Amplitudes

associated to these

diagrams

_are constructed in the

following

_way:

Vertices

,

, ,

,, ,

,, k, k' ,'

, ,

, ,

, ,

,, _, ,

, ,

, ,

' and ,'

a b _a ~

Fig. 3.

give respectively

< b Hi _a, k' _> and _< b, ^k' Hi a > the _saule is true if _oue of the Iine

a or b is doubled.

A flux of iutemal Iines

e~

Fig. 4.

gives

the factor

(Ef

+

( (E~,

+ hwk +

hwk,))~~,

if Iine e' is betweeu the externat

photons

vertices

(uot necessarily adjacent

to

theul).

It

gives

factor

(Ef

₊

( (E~,

+ hwk + hwk>

))~~,

if Iine e' is _on the Ieft of the ki

Photon

vertex or on the

right

^of the k2 _one.

The doubled Iine

gives

^the factor

(Ei

+

( R~(Ef

+

())~~.

So _one has

~~ÎÎ(f)

" C~e

~

^{< ei, k2 Hi e >}

~~ ~~~

~~ ~

~~ ÎÎ(ÎÎÎ ^~

^{~~ ~} _91[(e~

(t)

where

9111e~(f) =

/

_dk

_~ll

ll~' ~~ll~l~~~ii

^~

ll~ ll~

T(2)(~)

^Cte

£

^~

^{~2'~2 ~l}

^{~ > ~ ~ ~l}

^f'~l

^> _g(2)

~>~

Ef

+

(

E~ R~

(Ei

+

()

^~~'~~

where

~~~

l

j

^<

^{f Hi}

ei, k _> < ei, k Hi _e2 >

~~i'~~

Ef

_E~~ ^~~

(Ei

_E~~

hw)

^'

~Î~~~

^~~~

~ (EÎÎ~~ ~Î)Îifl _Î~~Î~~~ ÎÎÎ/ _~()

~~~~~~~~~~

where

< e Hi _ei, k _{> < ei,} k Hi _e2 >

9111e2

(f)

"

/

_dk

Ef

+

t Eei

h~

The

probability density

wiII

contain,

besides the

(2j)~

_{(2, the} crossed terms

Jl(T))~Î)$~ ).

^{We do}

not atm at _a

precise computation

of that

density; ôe

_{confine ourselves to}

a

brilf eiaminatiou

of _some ternis, the first _one for

instance, being

Ied then _to forecast the mentioned

discrepancy

for the exact

amplitude. Indeed,

in the

large wavelength approximation,

each term < ei, k2 Hi _{e2 >}

,

< _e Hi _f>

ki

> is of the form

C(ei, e2)(~i

So the function

~

~ ~~~

Îi _{~ÎÎ ~)~-} ÎÎ(ÎÎÎ ^Î~

^{~~ ~}

is

proportioual

to T$~~. Without the factor gÎ)Î~~

(~),

^{the shift of the}

peak

would

ouly

be due to the form of the

deuomiuator,

_{as we} said in

presentiug ApJi

But

£~~

~~

C(ei,e2)gÎÎÎe~

is uot _a constant; it is _a function

off, multiplyiug

the resouaut part ^{of the}

ainplitude,

and there

is no reason

why

the position ^of the

peak

should be Ieft

unchanged.

It must be

recognized

that the argument ^is

ouly

indicative.

Clearly

uot aII

diagrams

have beeu takeu iuto accouut,

and,

_eveu at _a

giveu order,

_ulany have beeu

ueglected. Iudeed,

the

terms T(~ ^{also shift the}

peaks,

aud the _sum of the

amplitudes

have to be

squared,

which

produces

crossed _terms. It follows from this that cancellatious of the different shifts could

occur. This cannot be excluded since _we lack of estimates of the

neglected

terms.

(Such

estimates _are dillicult since,

first,

the functions _{appear as} series

and, secondly,

the variable takes its value _{near a}

pote.)

However,

_we think the

preceding

overview _may

permit

to state the

following:

The real part of the

pote

of the Ti

amplitude

is

Iikely

to be different front the abscissa of the max1ulum of

the

peak

of the

probability density.

This

iu1pIies

that the real part ^{of the} resoIvant's

pote

is

Iikely

to be different from the _sum

f[

~

+

Ei.

Let _us put ^{it in the forul:}

~

In the

scattering

_~f + A _{- ~f} +

A,

with the

iucoming photon's

_euergy close to the differeuce of the

energies

^of _au ^excited state aud the fuudameutal _one, there is _no

simple

relation betweeu the abscissa of the maximum of the

peak

aud the real part ^{of the}

pote

of the Hamiltonian's resolvant.

3.4. CoNcLusIoN. As _a consequence, this real part ^is not

given directly by expenmental

data. It follows from this that excited _{states are}

only

accessible in _a mathematical way,

through

the

amplitude

and it's

potes. Only

the

profiles

of the

peaks

_can be tested

experimeutally.

4. Location of trie Peak in trie

Probability Density,

for 2-Photon Production

The fact still remains that _oue could define _{an energy} for the excited _state

by

the _sum of two

energies:

the fundamental _one and that of the maximum of the

peak.

For that defiuitiou to be

acceptable,

it is uecessary that this uumber should _not

depend

_ou ^{the final} state in

~f + A _- final state

(We

do not have here in mind the

dependence

of the transition matrix element _ou the outgoing test _wave

packet,

_{or on} the characteristics of the detector. We _are not either iuterested in the form of the incident _wave packet

[1].).

4.1. PROBLEM 2. CAN ONE DEFINE _A LEVEL ENERGY _BY

RESONANCES,

UNAMBIGU-

ousLY?. We _are

going

to show that the Iocalizatiou of the maxima of

peaks,

_lu the _cross- section for the

scatteriug

_~f + A _- final state, very

Iikely depeuds

_ou that final _state.

Whereas,

in Section

2,

_we cousidered the

probabilities

for

fiudiug photon

in the final state,

now we consider 2

photon

final states. The energy of the

iucoming photon

is still deuoted

by f.

The reductiou

procedure

indicated in reference _[3] for the expression of

<

Wi [H;nt, a2a3]

_W)~ >

gives:

T~ _{~_~ = a~} <

Wf ((Hint,a2],a31(£j

+

f H) _~(H;nt,

_a~Î

_i~f

_>

~+a2

_<

~f ÎlHint,a2j, aÎÎ(~Î ~3 ~) ~Î~lnt'~31 ~Î

~ +a3 <

ilf (ÎHint,a3j, aÎÎ(~Î

~2

~)

_~Î~Ànt'~21

~Î

~ +,~ <

#~ [Hint, a(](Ef f H)~~[[H;nt,

_{a21>a31 fl~f} > + +a~ <

§tf jH;nt>a3](Ef

+

h43 H)~~jjHint> ail,

£l21

1i~f

>

+a~ <

#f (Hint>a21(Ef

+ hw2

H) ~((Hjnt>aÎÎ,

_~31

i~f

> ~~~

+a7 <

~f ÎHjnt,a31(~Î

~

~~3 ~) ~Î~Ànt'~21~~Î

~

~~ ~Î~"~'~~Î ~Î

~ +aS <

~f ÎHint,a21(~Î

_~ ~~2

~) ~Î~Ànt'~31~~Î

~

~~ ~Î~"~'~~Î ~Î

~ +ag <

fllf jH;nt, a31(Ef

+ hw3

H)~~lH>nt, ail _(Ef ~1i~2 ~)~~l~int'~21 'f

~ +a10 <

~f ÎHjnt,a21(~Î

~

~2 ~) ~Î~Ànt'~il(~Î ~3 ~~ ~Î~"~'~~Î ~f

~

+OEll < it_f

[Hjnt, ~ll (EÎ f H)~~ ll"t'~31(EÎ à~~ H)~~ li"~'~~l

_~/~ ~ +OE12 < fit1

[H;nt, aÎÎ (Ei f H)

^~

(Hint, ~21(Ei

h~°3

H)

^~

ÎIÀ"t'~31 ~f

>

a~ are +1 _or -1.

We _are interested in the transition

probability density,

_{as a} function of

f,

alter summation

on alI

possible

states of the 2

outgoing photons.

We shall focus _on the

peaks

of that

function,

which _we deuote

by

_p2.

4.2. APPROXIMATIONS. As in the 1

photon

_case, it _seems uurealistic to expect

getting

the

position

of the

peaks

without

makiug

auy approximation, except in _{case one uses} numerical computations. ^So we shall

ouly

consider the Iast 6 terms in

(2),

and _more

specifically

the first of them. It exhibits

(the

second _one

also)

_{a resouauce} for f in _a

ueighbourhood

of E~

-Ef.

Indeed, [et _us Iook _at the Iowest

order, replaciug

ⁱⁿ

(2) ilf

>

by f,

fl _>

,

H

by Ho

and

keepiug only

the lowest order term

Hi

_m

H;nt,

_we introduce interulediate states

e', e", necessanly

photon

states.

(Summation

_on these states would have to be

performed

Iater _on

).

Theu _we get, for these 6 terms the

followiug diagrams (the

3

missiug

_are obtaiued

by permutiug

Iiues 2 aud 3:

"',

_(i)

_,1"

3

f e' e" f

,,"'

2 ,,""

',,, ^' (2) 2,,,' _--,,, ⁽³⁾ ,,,'

"",, _,,"' ""',,,

,,"" ₃

",, ,,' 3 ,,' j7>~Î, _,,," ,,,

',-f, ,,' _,,' ~',-~,,'

,,' _',, ,,' _,," ,,,' ~',,,,

f e' ^e" f f _e' e'~ f

Fig. 5.

Diagram

1

gives

the

pote considered,

if e'

= e.

(Diagram

2

might give

_a

singularity

at the _saule point, if e"

= e, due to the

siugulanty

at k2 _# 0 in the

integral.

At Iowest

order,

other ternis lu

(2) give diagrauls

^{with 4 Iine} vertices; the first _oue also has _a

pote

at the _same

point).

As in the 1

photon

_case, the

pote

of the

amplitude

is

expected

uot at

E~,

but at

E(;

this _is

seeu

by suululing

alI

diagrams

obtained from

diagraul

1

by inserting photon

Iiues _ou the Iine e' = e, as for

example

in the

followiug:

,--- ,--,, "~~',

' '~' _' '

,

' , , ^' ' ,

' , ' ' J ,

I ' '

',

_i

j

f ' t

e e

Fig. 6.

In order to discuss in _a heuristic way the position ^of the

peak

_near f ₌ f~, we shall cousider but the

followiug

terni of

(2), denotiug

it

by T[~_i.

T(~_~

= <

iii [H;nt, a31(Ei

+ hw3

H)~~[H;nt> a21(Ei

+

f H)~~[H;nt> a(] iii

>

In the _{saule way,} in Section

3,

_we

ouly kept T(_i.

As in

Paragraph 3.3, iutroducing

_{an expan-}

sion for

iii, together

with iutermediate states, we are Ied to such

diagrauls

as:

3 d"Î~' ~

,,

~'

'3,

2,'

,

, '

'

'

-~~~",

' '

'

,'

'W

'

', _i

~j~~g_1

,' ,,'

", ,,~'

,,'

',~,'

"

'~~""~'

'

"

'

"

'

,'

'

' ,'

V

' '

'

"

' '<

' "

,'

1

'

'

' '

'

' ' '

'

'

'

e

"

' ,'

'

~ f

"~ ~

~~ ~~ e~ _f

Fig. 7.

Let _us cousider the Ieft-hand _one,

aualogous

to

Diagram

1 of

Figure

2. The

correspoudiug amplitude

is the _{oue we} shall take _{as a}

model,

in the discussion of the

position

of the peaks ^of

the exact

au1pIitude.

_{ei, e2, e3 are}

supposed

to be

fixed,

the sumulation

being postponed.

4.3. THE PEAK. The

amplitude

associated to that

diagraulis:

~~~~°~ ~ ~~°~ ~~

~ ~

~~

_~~~~

~fÎÎ ~ÎÎ~~jÎÎÎEÎ~Î~~

^~~

^~~

^{~~ ~}

^~~~'~~~

where

~~~'~~~ Î _~~~

(Ei

+

f

hw

~Î~Î ÎÎI~ Î~Î~ ~ÎÎÎ-~Î~ ^ùJÎ Î(~f

hw E~~

Integrating

_ou k2 aud k3 with (k2( + (k3( #

(ki

^aud

summiug

_ou

polarizatious give

the

followiug

contribution jl2 to p21

fi2(~i~

₌

jE~

ÎÎÎ~ _Î~ ÎÎiÎÎ

t)j2

^~2~~~

with

f

=

h(ki(c

and

62(f)

#

dk2dk36(hùJ2

+ hùJ3

f)(x(f,ùJ2)(~ ~j

_<

_e21k3,

_{Àk~,n~ Hi}

e3 > (~

<0E2,0E3)E(1,2)2

.1 ~ e3i k2> Àk2,n2

1Hi

_{el > Î~}

The

problem

is to deteruliue the

peak

of that

deusity

jl2.

Actually,

there _are other

diagrams

to be considered and it is of _course the

peak

of the

cou1pIete probability density

that should be examined. Nevertheless, _as in Section 3, _we

just

wanted here _to show what kind of factors

might

shift the

peaks

from the

approxiulate

value

f~.

An

example

is the function 62. In Section 3, _an

analogous

term was, for

given

_ei and _e21

#1(ki)

= j~

)[' _(~

()~~(~~j~Ùi(f)

~~~~~

Ri

(f)

"

lgÎÎÎe2(f)Î~

~~~~~~~~ ~~~

~ ~~~

~ ~~'~~~_~~~~~

~~

_{~~ ~} ^~

4.4. COMPARISON WITH THE PEAK, IN THE 1 PHOTON CASE. We shall

simply

note that

the functions Ri et 62 _are not the _saule. It would be very

surprising

that exact computations,

takiug

account of alI the

diagrams,

and

performing

summations _ou iutermediate _states such _as ei, e2, e3 result in the _same

position

for the two

peaks.

We do uot _see auy reason for that. So

peaks

in 1

photon

_or 2

photon probabilities

_are

expected

at distinct

energies.

Let _us

crudely

schematize _our point in the

following

_way: the

righ-hand

_parts of

Diagrams

1 in

Figures

2 and 7, shift the

position

of the

peak

created

by

the denominator

(Ef

+

f

E~ R~

(El

₊

())

atone.

The two

corresponding

^shifts _{are a}

priori

^{different. We shall} state our

point

in the

following

form:

~ ^In the

neighbourhood

of _{a resonance,} the peak of the _cross section for the scat-

teriug

of _a

photon

_{on an} atom is _trot Iocated at the _same energy

accordiug

to whether 1 or 2

photons

_{are preseut} in the final state.

4.5. CoNcLusIoN _AND DIscussIoN. One _is then Ied to the conclusion that it does not _seem

possible

to

assign experimentally

_a ^real _energy to an atomic excited state, ⁱⁿ a

straightforward

way. It is of _course

possible

to

assign

a number to each

peak,

but what to do when the

theory gives

the

possibility

of 2

peaks

_near the _same value ? I must say I _am not competent to decide whether the location of the

peaks

for 1

photon

_or 2

photon

transitions _can be

experimentally distinguished,

with present

precisiou.

I thiuk that if there is _an

eventuality

that

they

_are

differeut from the theoretical point ^of

view,

it _must be cousidered. For the time

being,

it is diilicult to answer the

question:

_can

they

be

separated, by

calculus ?

Indeed,

_we often meutionned in the paper that the calculations

presented

here _were

only indicative;

_oue could expect ^that

good

estimates would be sullicieut to get ^the answer, but _{a receut} work _{ou a} toy model has showu _us that _accurate calculatious had uot

ouly

to take account of aII

diagrams,

but had also to coutrol the _sum of the series and to take account of the fact that there terms may have

potes.

An

important

fact that I think has beeu

ueglected

is that the shift operator ^matrix

elemeuts,

betweeu states of _{euergy say}

E,

have

potes;

this is due _to the fact that

particular

intermediate states, ^{in the} expansion of such matrix elemeuts in powers of the

coupliug

constant, _may be atomic states of _energy

E',

with _no

photons.

These

potes

make estimates of the _sum of the series _{a non} trivial task.

In _a

pecular model,

in which calculations _cau be

performed completely,

_{one sees} that the answer, even

qualitatively, depends

_on these delicate

questions

of

singularities

and convergence;

the corrections to the first estimates of the

shifts,

when

they

_are

accurately calculated,

_are uot of

uegligable

order. This Iets _us thiuk that in a realistic atomic

model,

the precise location of the

potes,

or of the

peaks,

_is ^diilicult to get. ^{It is} uot _eveu clear that to each

unperturbed

state

corresponds ouly

_oue pote. In the model _we

just spoke about,

it is

precisely

not the _case.

We _must uuderliue that this

geuerally

admitted

correspoudence

has _trot

been,

to _our knowl-

edge,

^confirmed

by

theoretical

calculations,

and the present

study gives,

in _our

opinion,

_reasous

to think that it

might

be false. At the moment, ^the theoretical

aualysis

is uot advanced

euough

to state that

quautitatively.

It is the aim of the _{paper to} mtroduce the question aud to

give

motivations for future

analysis allowing

_progresses in that direction. It suggests ^that a

rigorous

mathematical treatment of the

perturbation

of

eigeuvalues,

in such _a type ^of interaction _as the atom-radiation _oue, would be of interest; ^the

important

features of the models should

be,

_as in the

physical

situation, the iufinite uumber of the

particles (photons)

and the

degeueracy

of the Ievels

(of

the eutire

system).

It is _oue of the conclusions _{one may} draw from _our

study,

that

the one-to-one

correspondence

between Ievels aud

potes, generally

admitted, is not evideut.

Let _us recall the 2 main arguments: ou one hand, there is

Iikely

_a

discrepancy

between the abscissa of the maximum of the

peak

and the dilference of the real parts of the

correspondiug

assumed

potes;

_on the other

haud,

the _resonauce

energies

^for _a

photon impinging

_{on an} atom in its fundamental state

might depend

_on the final states.

Appendix

A

Let _us show how _one obtains

Diagram

1 of

Figure

2 from

T(_~.

Replaciug,

in

T(_i,

the resolvant G

by

its

expansion

_up to the first

order,

G

=

Go

+

GoHi Go,

one gets:

Tl

_Ci <

i~i [Hi,a~lGolHi,ail i~i

> + <

i~i [Hi,a~lGoHiGolHi, ail i~i

>

In the second term, [et _us introduce _a summatiou _on intermediate states; it

produces

terms:

£

_<

_iii [Hi a2]Go(Ei

+

()Hi

_{e > < e}

[Hi, a(] iii

>

(Ei

+

( E~)~~

e

Let _us

replace

the vector

iii

at the

right-hand

side

by f

> and that at the Ieft _oue

by

its

expansion

_up to the first order:

<ilf(

₌ <

fi +~j /dk ~~~~~~~~'~~

<ei,k(

~~

Ef Eei

~

(summation

_on

polarizations

_are included in

f).

In the term < ei, k

[Hi,a2]GoHi

_{e >} thus

obtaiued,

[et _us agaiu introduce _a summation

ou iutermediate states; those _are

necessarily

1

photon

states, ^with wave uumber

k,

because of the scalar nature of

[Hi,

_a2]

operatiug

_ou

7iradi

_we deuote them

by

_e2, k > We thus get, for _a

particular

_e:

£

_dk<

f(Hi(ei,k> <ei,k2(Hi(e2> <e2,k(Hi (e> <e(Hi(f,ki>.P(f)

i,e2~

where

P(f)

"

(EÎ

+

f Ee)~~(Ef Eei

hW)~~

(EÎ

+

f Ee2 à~)~~

One

recognizes

the

amplitude

associated to the

diagram considered, usiug

mies

given

in Para-

graph

3.3.2.

References

iii

Arnous E. and Heitler W., Froc. Roy. Soc. A 220

(1953)

290.

(2] Cohen-Tannoudji C., Dupont-Roc J. and Grynberg G., Photons et atomes. Introduction à

l'électrodynamique quantique

(Interéditions/Editions

du C.N.R.S., Paris,

1987);

english trans- lation

(J.

Wiley, New York,

1989).

(3] ^{Kroll N.} M., "Quantum Theory of Radiation", Quantum Optics and Electromcs, Les Houches

(1964),

C. Witt, A. Blandin and C.

Cohen-Tannoudji,

Eds.

(Cordon

and Breach, New York,

1965),

_p. 1.

(4] ^{Power E. A. and Zienau} S., Phil. Trans. A 251

(1959)

427.

(Si Prigogine I. and Petrosky T., Classical and Quantum Systems; Foundations and Symmetries,

H-D- Doebner, ^W. Scherer and F. Schroeck, Jr., Eds.

(World

Scientific,

1993).