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Prog. Theor. Phys. Vol. 91, No.6, June 1994, Letters
The New Decay Mode in the Electric Dipole Transitions
--Inversely Proportional to the Eighth Power of Time-Gi-Chol CHO, Hikoya KASARI and Yoshio YAMAGUCHI
Department of Physics, Tokai University, Hiratuka 259-12
(Received March 28, 1994)
We investigate the part of the electric dipole transition, whose time dependence is inversely
proportional to the 8th power of time in the leading order. To be specific, we examine the electric
dipole transition (2p->1s) of the hydrogen atom and the hydrogen-like atoms. The results are of
academic interest only, since they are so small to propose any experimental test at the moment.
In previous papers, we have presented the general theory of quantum mechanical
resonances 1> and discovered a new decay mode (with dependence on a certain inverse
power of time) of unstable particles,2 > apart from the well-known and overwhelmingly
dominant exponential decay mode. In this paper, we shall discuss explicitly such a
decay mode associated with the electric dipole transition (2p~ ls) of the hydrogen
atom and other hydrogen-like atoms (namely, systems consisting of one electron plus
a nucleus). To this end, we shall begin with the brief summary of our argument
developed in the previous paper, 2> modified to the case of electric dipole transitions.
Let us take two channels, P- and D-channel to be
P-channel: one H-atom with no photons,
D-channel: one H-atom with one photon.
We are using the Fock space representation.
in these two channels by HP and Hn.
Then we see that
Let us denote the (free) Hamiltonians
where His the Hamiltonian for the hydrogen atom, aMakE) is the creation (annihilation) operator of a photon with momentum k and polarization E, and
w=\k\.
The channel coupling Hw is the electric dipole operator:
Hw=ie/4i ~ (er)ak.+h.c .
.;2w
Now, we can write down the basic Shrodinger equation for our problem:
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Letters
Vol. 91, No.6
(1)
For the 2P->1s transition, we should take Q to be
Q=-E(1s).
We denote the 2P wave function (wave function in the P-channel) by <PB and the
"continuum" state wave function in the D-channel by ¢k (following the notation used
in Ref. 2)):
(2)
(3)
or more explicitly
r/>kiO>=R10 Yooa~.IO>,
E(1s)=-
~ a2me, E(2p)=- ~ a2me.
From these equations we can derive the survival probability amplitude of the 2P-state
(at rest) of the H atom prepared at time t=O
A(t)=e-i(Eo-(i!2)ro)t
-1:"" (~:~3 I< WkiHwi<PB>I2e-iEkt'
(4)
and the survival probability:
P(t)=IA(t)IZ
(5)
where Eo and I0 are the resonance energy (i.e., Eo=E(2p)- E(1s)=co) and its width,
respectively. n is a certain number which depends on the nature of final states (n=4
for dipoletransitions, see Eq. (13)). The coefficients T1, T2, ··· stem from the Taylor
expansion of I< WIHwl <PB>I 2 at E=O, i.e.,
7}1
(Ji.)n
t -{1<
-
11r
'1' k
1m
'VB >12} Ek=O
1°
-ioo
2
dk d'Eke -iEkt ,
47rk dEk
(2Jr)3
1
T2 )n+ -{...!LI< 7Tr 1m >12}
2
(
- dE '1' k 'VB
7} t
Ek=O
1°
2
47rk ___!jfi_d'E -iEkt
-ioo (2Jr)3 dEk
ke
'
(6)
(7)
where phase factors 7}1, 7}2, ···(I7J1I=I7J21=···=1) have been introduced to make T1, T2,
··· real (and positive) and
T1, T2, ··· can be evaluated by making use of perturbation theory, e.g.,
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June 1994
r;1( T1)n =1° 4Jrk:
-ioo (2Jr)
f
2
_dk_dEki<if!kiHwi(])B>i e-iEkt
E-(Q- B) E=O
dEk
-1° 4Jrk _dk_dE l<if!kiHwi(])B)I~=o -iEkt
'
e
(Q- B)2
k
-ioo (2Jr)3 dEk
2
(8)
and so on. We shall retain only the leading term (T1/t) below because the higher
order terms in Eq. (5) are very small as discussed in Ref. 2). Equations (4) and (5) are
valid for "large" t( r < 1/I'o).
It is interesting to calculate the critical time t*, at which
e
-(t*/<o)
=
(Il)
t*
2
n
(9)
holds, where ro denotes the mean life (ro=1/I'o).
First, we write down the transition probability per unit time:
(10)
where l<f1Hwli>l 2 is square of the transition matrix element between the initial state
i and the final state f, Ii> ~If>,
2
l<f1Hwli>l 2 =liefu ~ <iiErlf>l
v2cv
=2Jracvl<ilerlf>l 2 ,
(11)
where
E:
polarization vector of emitted photon ,
E;: initial energy of hydrogen atom ,
Ef: final energy of hydrogen atom,
and a=e 2=1/137(n=c=1). The transition probability per unit time of the 2p~1s
transition of the hydrogen atom is given by
= 1.60 X 10- 19 sec.
In this case, we used the initial and final state as follows:
li>=R21 Yio,
If>= R1o Yoo .
Next, using Eq. (8),
2
e-iEkt
-1° 4Jrk: _dk_dEkl (if!kiHwi(])B) 1
Ek=O
B)
E-(QdEk
-ioo (2Jr)
(12)
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Letters
Vol. 91, No.6
=-(~1r,
(13)
we find
(14)
namely n=4 and 1]1 = -1. Note that the emitted photon in the final state has "zero"
energy w ~ 0.
The survival probability P(t) for macroscopic time is
(15)
The leading term of inverse power part is given by ( Tdt) 8 , i.e., inversely proportional
to the 8th power of t. The critical time t*, defined by Eq. (9) can be found from
I'ot* -8lni'ot*= -8lni'o T1.
(16)
t* for the 2P--" 1s transition of the hydrogen atom is
t*~220ro.
(17)
Finally, we can extend our result to
the electric dipole transition of any
hydrogen-like atom (one electron
+nucleus of atomic number Z), whose
I'o(Z) and T1(Z) are related to those (I'o,
T1) of the hydrogen atom by the following scaling laws:
I'o(Z)=Z 4 I'o,
( T1(Z)) 4 =
i6
r,(Z)t'- Bini'o(Z)t'
--~~~-.~~~~.-~~~
200
Z=1
Z= 10
100
z
T1 4 ,
z
z =30
=50
= 92
so that
0
zs/2
0
100
200
t'
r,(z)t' =To
Consequently, Eq. (16) will be modified
to
I'o(Z) t*- 8lni'o(Z) t*
=-8lni'o(Z)T1(Z)-20lnZ. (18)
The numerical results for cases Z=1, 10,
30, 50 and Z=92 are depicted in Fig. 1.
Fig. 1. The figure depicts the critical time t*
defined by Eq. (18), To(Z)t*-8JnTo(Z)t*=-8ln
To TI-20lnZ for the hydrogen-like atoms in the
electric dipole transition 2P-> 1s, whose transi·
tion probability per unit time is To(ro=1/To).
The solid line shows the quantity Fo(Z)t*-8Jn
Fo(Z)t* vs Fo(Z)t*. We marked here for sev·
era! choices of atomic numbers Z=1, 10, 30, 50
and Z=92.
June 1994
Letters
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In general, the index n of the inverse power mode of survival amplitude for the
electric or magnetic Lth multipole transition is given by 2L+2, since the transition
2
matrix element squared I<!!Hwli>l is proportional to (L-1/2)th power of energy.
This means that the survival probability contains the leading inverse power term
(1/t) 4L+\ so the critical time t* increases with L.
It is also easy to derive the part which depends on the inverse power of time for
the radiative transition in atomic nuclei. In the case of both atomic and nuclear
radiative transitions, the critical time t* is too large to verify the existence of the part
with inverse power law.
1)
2)
G. C. Cho, H. Kasari andY. Yamaguchi, Prog. Theor. Phys. 90 (1993), 783.
G. C. Cho, H. Kasari and Y. Yamaguchi, Prog. Theor. Phys. 90 (1993), 803.