J. Mod. Phys., 2010, 1, 290-294
doi:10.4236/jmp.2010.14040 Published Online October 2010 (http://www.SciRP.org/journal/jmp)
Copyright © 2010 SciRes. JMP
Quantum Theor y of a Ra diat ing Harmonically
Bound Charge
Emilio Fiordilino
CNISM and Diparti mento di Scienze Fisich e e Astron o mic he Universi t à di Pale rm o,
Via Archirafi36, 90123Palermo, Italy
E-mail: fiordili@fisica.unipa.it
Received July 8, 2010; revised August 6, 2010; acce pted August 18, 2010
Abstract
A phenomenological Hamiltonian giving the equation of motion of a non relativistic charges that accelerates
and radiates is quantized. The theory is applied to the harmonic oscillator. To derive the decay time the
physical parameters entering the calculations are obtained from the theory of the hydrogen atom; the agree-
ment of the predicted value with the experiments is striking although the mathematics is very simple.
Keywords: Radiation Damping, Quantum Radiation, Phenomenological Hamiltonian
1. Introduction
A classical particle with charge q and mass m, mov-
ing non relativistically with acceleration ()ta, emits
electromagnetic radiation whose power is given by the
Larmor formula:
 
22
3
2
=.
3
q
Pt t
ca (1)
The equation of motion of the charge is modified by
the loss of energy and assumes a form that is an open
question and object of debate. The traditional non rela-
tivistic equation of motion for the charge moving in the
presence of an external force (,)tFr has the non new-
tonian form:

23
23
=,
dd
mtm
dt dt
rr
Fr (2)
where 23
=2 /(3)qmc
is 2/3 of the time taken by
the light to travel trough th e classical radius of the charge ;
for the electron its value is 23
=0.6310
sec. The
equation of motion (2) was already known to H. A. Lor-
entz at the beginning of the XX century but it was P. A.
M. Dirac [1] who gave a relativistically invariant method
to derive the radiation reaction of which Equation (2) is
the non-relativistic limit. A different approach taking to
the same equation can be found in [2].
The standard solution of (2) requires three initial con-
ditions and presents many problems of interpretation; if
the force is function of the time alone, the acceleration
obtained by solving Equation (2) is:
20
0
2
111
0
=exp( )exp
1exp( )
t
t
dt t
tt
dt
ttdt
m



 
 




ra
F
(3)
here 0
()ta is an integration constant which one is natu-
rally pushed to identify with the initial acceleration at
time 0
t. Unfortunately by doing so ()tr
 diverges for
large values of the time even when the external force has
been switched off. Dirac [1] showed that the difficulty
can be circumvented by requiring that the final accelera-
tion is zero. By following this idea the solution of the
equation of the motion is:
 
()/
111
1
,= ,
tt
t
te tdt
m
arFr (4)
which does not diverge but causes the particle to interact
with the future; the final effect is that the charge begins
to accelerate the tiny time
before the onset of the
force [3].
Exploiting the fact that
is very small, it is possible
to substitute in Equation (2) the quantity (,)tFr
to
()mta
; in this way the equation of motion becomes

==
d
mdt t





FF
aFFv F
(5)
that has a newtonian form and does not present exotic
solutions; it can be considered an approximation of (2)
E. FIORDILINO
Copyright © 2010 SciRes. JMP
291
for
||.
d
dt
FF
(6)
Equation (5) has been considered by Rohrlich in [4] to
discuss the secular question of the radiation of an uni-
formly accelerated charge. Di Piazza in [5] has used its
relativistic generalization to find the motion of a charge
in the presence of a plane electromagnetic wave.
Let us consider now a charge acted upon by a one di-
mensional harmonic force
F
with free angular fre-
quency so that 2
=
F
mx ; in this case Equation
(5) gives the equation of a damped harmonic oscillator
with damping constant 2
=
:
22
=0xxx
 
  (7)
with solution
 
= expsincos
2sc
x
txt xt







(8)
where
x
and c
x
are integration constants and
22
2
=48
 
(9)
is the angular frequency of oscillation of the damped
oscillator and shows that the free angular frequency is
shifted of the quantity 23
=/8
 .
It is not surprising that the theory of a radiating charge
is much better understood in the quantum realm where
the radiation is the result of spontaneous transitions be-
tween stationary states induced by the interaction of the
charge with the vacuum radiation field and where the
frequency Lamb shift results from emission and absorp-
tion of a photon by the oscillator [6,7]. At the moment
the quantum electrodynamics is the theory that best
agrees with experiments and the only coherent way to
treat the radiation from charges; the price to be payed is
that the theory is not easily applied so that physical
problems are usually modelled by forgetting completely
the radiation reaction even when this should be present at
least in principle. Generally speaking this is a good as-
sumption, however .
Recently the development of high power lasers opened
the channel to so called laser induced effects. Among
these the process of high order harmonic generation is
relevant for our discussion. Here atoms in the presence
of a laser field of frequency
L
become source of new
electromagnetic radiation whose spectrum is formed by a
large plateau of odd harmonics of
L
. The maximum
detected harmonic order can be as large as 300 [8]. The
radiation origins from the strong acceleration induced by
the laser upon the charges of the atom. By assuming that
only one electron is active, the emitted power is given by
the Larmor formula with ()ta the quantum averaged
electron acceleration:
  
2
2
=,,
d
ttt
dt

arrr
(10)
and (,)t
r the wave function of the electron in the
presence of the laser. Here again the radiative reaction is
never taken into account even in principle.
This paper approaches the problem of the quantum
theory of a radiating charge from a phenomenological
point of view; the lack of first principles ground of the
treatment will be justified by the simplicity of the
mathematical machinery and by the results. The kernel of
the idea resides in the quantization of a hamiltonian that
gives the correct form of the classical equation of motion.
Unfortunately there is not any hamiltonian that gives
Equation (2) but it is possible to construct one for Equa-
tion (7). This paper applies the idea and shows that the
method provides correct quantitative predictions.
2. Theory
The Hamilton equations applied to the classical hamilto-
nian:
222
1
=22
tt
p
H
eemx
m

 (11)
give the expressions:
2
==
t
H
pex
x

(12)
for the time derivative of the canonical momentum and
==
t
H
p
xe
pm
(13)
for the velocity; the two combined expressions give the
equation of motion of a damped harmonic oscillator:
2=0;xx x

  (14)
after comparison with Equation (7) we obtain:
2
=.

(15)
From the expression for the velocity (13) we see that
the kinetic momentum mx
differs from the canonical
momentum p: the quantity 2/2pm
cannot be given
the meaning of kinetic energy; indeed the expression for
the total mechanical energy is:

222
11
==
22
t
Emx mxeH

(16)
showing that
H
cannot be identified with the energy of
the oscillator. This is not new in the domain of electro-
dynamics where the Hamiltonian of a charge q interact-
E. FIORDILINO
Copyright © 2010 SciRes. JMP
292
ing with an electromagnetic field with potentials (,)tAr
and (,)tr is
 
2
1
=,,
2
qtt
mc




pAr r (17)
giving for the kinetic momentum =(/)mqcrp A
so
that 2/2mp is not the kinetic energy.
Qu antu m mec han ics has a general quantization recipe:
in the hamiltonian substitute classical quantities Q with
operators ˆ
Q; therefore the hamiltonian (11) assumes a
quantum role after the substitutions ˆ
H
H and
ˆ=
x
ppi.
The exact solution of the related time dependent
Schroedinger equation
 
,ˆ
=,
xt
iHxt
t
(18)
is thoroughly derived in [9] and subsequently applied to
the problem of the coupling of a spin to the modes of a
cavity [10]; the interested reader is therefore referred to
these works. The important point is that the exact quan-
tum solution displays an oscillating solution with free
angular frequency given precisely by
; therefore an
equivalent of the Lamb shift is obtained also within this
phenomenological treatment.
Here, to explore the potentiality of the method, we are
interested in an approximated solution which exploits the
smallness of the damping factor. If
222
0
ˆ1
ˆ=22
p
H
mx
m
(19)
denotes the formal quantum hamiltonian of the non radi-
ating oscillator the hamiltonian (11) can be written as
22 22
0
22
0
11
ˆˆ
=22
ˆ
=sinh().
tt
t
H
eH mxemx
eH mtx


 



(20)
Again neither ˆ
H
nor 0
ˆ
H
can be considered energy
operator since p
ˆ is not the kinetic momentum.
Since
is small we can Taylor expand ˆ
H
:
(sinh( )1tt
)

242
0
ˆˆ
1
H
tH mtx
 (21)
and solve the Schroedinger equation with 2
=
as
perturbation pa rameter. We set:
 
()
=0
,= ig t
n
n
n
x
taen
(22)
and substitute it in the time dependent Schroedinger
equation. Usual projection procedure gives:




()
()
222
()
42 2
2
,2
()
42 2
2
,2
=
ig t
ig t
ig t
ig t
ia igae
mx tae
mtx ae
mtx ae
 

 










(23)
with 01
ˆ|)=|)=() |)
2
n
H
nnnn
 and

 
22
,2
,,2
1
=2
12
1;
22
kn
kn
knkn
xkxn
nn
m
nn
n






(24)
we stress that n
cannot be interpreted as the energy
of the eigenstate |)n since 0
ˆ
H
is not the energy op-
erator. The system of equations in (23) is readily solved:

1
=2
g
tt




(25)
and


42 2
2
,2
22
2
,2
=
.
it
it
iamtxa e
xae




(26)
The system is solved with the initial condition
,
(=0)=
K
at a

. First order perturbation theory re-
duces the system of equations to:
 

3
(0) 2
2
3
(0) 2
2
=12
2
=1
2
it
K
it
K
iaKKe t
iaKKe t


(27)
which after integration gives:
 

2
(0) 2
2
2
(0) 2
2
=12
4
sin
=1
4
sin
K
it
it
K
it
it
aKK
e
te t
aKK
e
te t



 


 

(28)
correct for 2
1
t
.
E. FIORDILINO
Copyright © 2010 SciRes. JMP
293
2.1. Estimations
The transition probabilities are now worked out:
 


22
2
(0) 24
22 2
=12
2
sin 2
sin
K
aKK
tttt

 
(29)



22
2
(0) 24
22 2
=1
2
sin 2
sin
K
aKK
tttt
 
(30)
having in mind the hydrogen atom in the first excited
state we set

4
21 33
13
==.
2
me
EE
(31)
In Equation (29) the dominant term is the first in curl
brackets. By defining the time T such that
(0) 2
2
|()|=1/2
K
aT
, we have:
242
3
63 9
34
1
=2
2
2
==1.33 10sec
3
T
Tme

(32)
in striking agreement with the experimental value of
9
=1.610
exp
T
sec. It must be stressed now that the
value of the shift frequency  defined in Equation (9)
and recovered by the exact quantum treatment [9] is very
much smaller than the measured Lamb shift.
Now we work out the time dependent average me-
chanical energy of the system. Since the particle starts
from the state K we have
  

12
2
,=1 2sin2
4
1sin2,
it
iK t
K
it
e
xteKKKtt K
e
KKt tK
 





 


 


(33)


 

 
1
2
2
2
12
42 2
1
ˆ,= 12
1
12sin 22
42
1
1sin22
2
12 sin
4
iK t
K
it
it
it
iKt
HxtetK K
e
KK ttKK
e
KKttK K
e
mtxeKK KttK






 




 











 





 



2
1sin2.
it
e
KKt tK


 



(34)
To first order in
:
 

*242
,
11 1
ˆˆ
,,== .
22 2
KK
KK
HxtHxtdxK tKmtxK
 





  (35)
Our previous discussion on the energy of the radiating
oscillator (16) shows that the energy decays as
ˆˆ
()=(1)
t
EteHtH
  as it should be.
3. Discussion
Hamiltonians of the type of formula (11) have been in
the past applied to study the quantum counterpart of a
particle in the presence of a mechanical dissipative force;
the conclusion has often been negative: the quantum the-
ory of particles moving in the presence of dissipative
force is not yet complete. In [11] furthermore the very
basic meaning of the mechanical system described by the
hamiltonian (11) is questioned. In [12,13] an approach
very similar to the one outlined in this work has been
presented with purpose to study the somewhat artificial
problem of quantum particles in the presence of a friction.
The present work considers the radiation process as cre-
ating a dissipative medium and finds its frame in the
domain of phenomenological quantum electrodynamics
E. FIORDILINO
Copyright © 2010 SciRes. JMP
294
and not in the realm of mechanics.
The theory here outlined gives a practical, albeit ad
hoc, method of incorporating the radiation reaction in the
quantum treatment of a harmonically bound charge. Re-
cently [14] it has been shown that the nuclei of a dia-
tomic molecule driven by a laser field can be profitably
described by assuming that they are bound by a harmonic
potential so that such a system might provide a bench-
mark for the theory here presented.
4. Acknowledgements
This work has been supported by the Italian Ministero
dell'Istruzione, dell'Università e della Ricerca and has
been outlined while visiting A. Maquet in Paris. Discus-
sions with A. Di Piazza, M. H. Mittleman and F. Persico
are gratefully acknowledged.
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