Long-range interactions of hydrogen atoms in excited states. I. 2S-1S interactions and Dirac-delta perturbations

0.89 MB·

by  C. M. Adhikari

Long-range interactions of hydrogen atoms in excited states. I.
2S–1S interactions and Dirac–δ perturbations
C. M. Adhikari,1 V. Debierre,1 A. Matveev,2,3 N. Kolachevsky,2,3,4 and U. D. Jentschura1
1Department of Physics, Missouri University of Science and Technology, Rolla, Missouri 65409-0640, USA
2P. N. Lebedev Physics Institute, Leninsky prosp. 53, Moscow, 119991 Russia
3Max–Planck–Institut fu¨r Quantenoptik, Hans–Kopfermann-Straße 1, 85748 Garching, Germany
4Russian Quantum Center, Business-center “Ural”, 100A Novaya street, Skolkovo, Moscow, 143025 Russia
7 The theory of the long-range interaction of metastable excited atomic states with ground-state
1 atoms is analyzed. We show that the long-range interaction is essentially modified when quasi-
0 degenerate states are available for virtual transitions. A discrepancy in the literature regarding
2 the van der Waals coefficient C6(2S;1S) describing the interaction of metastable atomic hydrogen
(2S state) with a ground-state hydrogen atom is resolved. In the the van der Waals range a ≪
n 0
R≪ a /α, where a = ~/(αmc) is the Bohr radius and α is the fine structure constant, one finds
a 0 0
J thesymmetry-dependentresultE2S;1S(R)≈(−176.75±27.98)Eh(a0/R)6 (Eh denotestheHartree
0 energy). In the Casimir–Polder range a0/α ≪ R ≪ ~c/L, where L ≡ E(cid:0)2S1/2(cid:1)−E(cid:0)2P1/2(cid:1) is
theLamb shift energy, one findsE (R)≈(−121.50±46.61)E (a /R)6. In the theLamb shift
3 2S;1S h 0
range R ≫ ~c/L, we find an oscillatory tail with a negligible interaction energy below 10−36Hz.

] Dirac–δ perturbations to the interaction are also evaluated and results are given for all asymptotic
h distance ranges; these effects describe the hyperfine modification of the interaction, or, expressed
p differently, the shift of the hydrogen 2S hyperfine frequency due to interactions with neighboring
- 1S atoms. The2S hyperfinefrequencyhasrecentlybeenmeasuredveryaccurately inatomicbeam
m
experiments.
o
t PACSnumbers: 31.30.jh,31.30.J-,31.30.jf
a
.
s
c
I. INTRODUCTION ifications of the long-range interactions result from the
i
s presence of the quasi-degenerate states.
y
h The purpose of this paper is twofold. First, we aim to Recently, precision measurements of the 2S hyperfine
p revisit the calculation of the long-range (van der Waals splittinghavebeencarriedoutusinganatomicbeamcon-
[ and Casimir–Polderinteraction) for ground-state hydro- sisting of a mixture of ground-state 1S hydrogen atoms,

geninteractingwithanexcited-stateatominan2S state. andmetastable2Satoms[6,7]. Toleadingorder,thevan
1
v Second, we aim to study the perturbation of the van der Waals interaction shifts all hyperfine structure com-
2 der Waals interactions by a Dirac-δ potential perturb- ponentsequally. However,thereisacorrectiontothevan
1 ing the metastable excited state which participates in der Waals interaction due to the the hyperfine structure
8 the interaction. Such a Dirac-δ potential can be due to (HFS), which depends on the total (electron+nucleus)
8 the electron-nucleus (hyperfine) interactionin one of the angularmomentumquantumnumberF. Thiscorrection
0
atoms [1] or due to a self-energy radiative correction [2]. shifts HFS components closer to each other. This latter
.
1 Special emphasis is laid on the role of quasi-degenerate effectisanalyzedinthecurrentpaper;itisofphenomeno-
0 levelsandontheexchangeterm,whichisduetothepos- logicalsignificancebecauseofvander Waalsinteractions
7 sibility of 1S–2S atoms becoming a 2S–1S pair after the inside the atomic beam. Again, special attention is re-
1
exchange of two virtual photons [3]. quired in the treatment of the quasi-degenerate atomic
:
v levels.
It is interesting to notice that two results given in the
i
X literature for the so-called van der Waals coefficient of Letusrecallherethatthegeneralsubjectoflong-range
the 1/R6 nonretarded interaction between 1S and 2S interactions of simple atoms is very well known to the
r
a states, are in significant mutual disagreement (numeri- physics community, and a few investigations on simple
cally, the authors of Refs. [4] obtain a value of roughly atomic systems can be found in Refs. [4, 5, 8–17]. Var-
177 in atomic units, while a result of about 57 has been ious aspects of the problem have been studied in depth:
derived in Refs. [3, 5]). We attempt a thorough analysis e.g., the importance of multipole mixing effects, and of
of the discrepancy. Two different methods of calculation perturbations by hyperfine effects, has been stressed in
wereemployedinRef.[4](directsumovervirtualatomic Ref.[10–13]. Higher-ordereffectssuchasdipole-octupole
states, including the continuum) and Refs. [3, 5] (inte- mixing terms were discussed in detail for hydrogen in
gration over analytic expressions representing the polar- Ref.[5]andforheliuminRef.[16]. The dipole-dipolein-
izability). teraction potential of helium, including retardation, has
The role ofthe virtual, quasi-degenerate2P states de- been discussed in great detail in Refs. [18, 19], including
serves special attention. For 2S reference states, the a number of numerical examples. More complex alkali-
2P and 2P levels are displaced only by the Lamb metal dimers have been considered in Refs. [20, 21].
1/2 3/2
shift and fine structure, respectively. Significant mod- Throughoutthisarticle,weworkinSImksAunitsand
2
keep all factors of ~ and c in the formulas. With this wheremistheelectronmass,the~p denotethemomenta
i
choice,weattempttoenhancetheaccessibilityofthepre- ofthetwoatomicelectronsrelativetotheirnuclei(iruns
sentationtotwodifferentcommunities,namely,thequan- over the atoms A and B), and the ~r = ~x R~ denote
i i i
tumelectrodynamics(QED)communitywhichingeneral the coordinates relative to the nuclei (the e−lectron and
usesthenaturalunitsystem,andtheatomicphysicscom- nucleus coordinates are ~x and R~ , respectively). We re-
i i
munity where the atomic unit system is canonically em-
strictthe discussionto neutralhydrogenatoms andthus
ployed. In the former, one sets ~ = c = ǫ = 1, and the
0 assume a nuclear charge number of Z =1. We shall use
electron mass is denoted as m. The relation e2 = 4πα
thefollowingapproximationtothe“LambshiftHamilto-
then allows to identify the expansion in the number of
nian”, which constitutes an effective Hamiltonian useful
quantum electrodynamic corrections with powers of the
in the evaluation of the leading radiative correction to
fine structure constant α 1/137.036. This unit system
dynamic processes [24, 25],
≈
is used, e.g., in the investigation reported in Ref. [22] on
relativistic corrections to the Casimir–Polderinteraction 4 ~ 3
(withastrongoverlapwithQED).Intheatomicunitsys- HLS = α2mc2 ln α−2 δ(3)(~ri) . (3)
3 mc
tem, we have |e|=~=m=1, and 4πǫ0 =1. The speed i=XA,B (cid:18) (cid:19) (cid:0) (cid:1)
oflight,intheatomicunitsystem,isc=1/α 137.036.
≈ We shall use this Hamiltonian later in the analysis of
This system of units is especially useful for the analy-
the radiativecorrectiontothe long-rangeinteratomicin-
sis of purely atomic properties without radiative effects.
teraction. The Hamiltonian for the hyperfine interac-
As the subject of the current study lies in between the
tion [1, 26] reads as
two mentioned fields of interest, we choose the SI unit
system as the most appropriate reference frame for our
µ 8π
calculations. The formulas do not become unnecessarily H = 0 µ µ g g S~ I~ δ(3)(~r )
HFS B N s p i i i
4π 3 ·
complex, and can be evaluated with ease for any experi- i=A,B(cid:20)
X
mental application.
3(S~ rˆ)(I~ rˆ) (S~ I~) L~ ~µ
We organize this paper as follows. The problem is + i· i i· i − i· i + i· i . (4)
somewhat involved, as such, we attempt to orient our- ~ri 3 ~~ri 3 #
| | | |
selves in Sec. II. The direct term in the 2S–1S interac-
tion is analyzed in Sec. III. In Sec. IIIB, we study that Here,theunitvectorsarerˆi =~ri/~ri . Thespinoperator
| |
interaction in the van der Waals range. The very-large- for the electroni is S~ =~σ /2,while I~ is the spin opera-
i i i
distance limit is discussed in Sec. IIIC (atomic distance tor for proton i (both spin operators are dimensionless).
largerthanthe wavelengthofthe Lambshifttransition), The electronic and protonic g factors are g 2.002319
s
andthe intermediate Casimir–PolderrangeinSec. IIID. and g 5.585695, while µ 9.274010 ≃10−24Am2
p B
≃ ≃ ×
The mixing term in the 2S–1S interaction is analyzed is the Bohr magneton and µ 5.050784 10−27Am2
N
in Sec. IV. We then analyze a Dirac-δ (HFS induced) is the nuclear magneton [27]. I≃t is well kno×wn that, for
induced modification both for the 2S–1S interaction as S states, the second term in the fine structure Hamilto-
well as for the 1S–1S interaction in Sec. V. In Sec. VI, nian(2),andthesecondandthirdtermsinthehyperfine
wenumericallyevaluatethe shiftofthe2S hyperfinefre- structure Hamiltonian (4) have vanishing contributions.
quency due to the long-range interaction with a ground For S states, the relevant term in the hyperfine Hamil-
statehydrogenatom. ConclusionsaredrawninSec.VII. tonian therefore is of the Dirac-δ type. Hence, we put
specialemphasisonthemodificationsoccasionedbysuch
Dirac-δ potentials.
II. ORIENTATION The van der Waals energy is normally derived as fol-
lows. One first writes the attractive and repulsive terms
In order to evaluate the van der Waals correction to that describe the electron-electron, electron-proton, and
the 2S–1S hyperfine frequency, one needs to diagonalize proton-proton interactions in the two atoms (excluding
the total Hamiltonian theintra-atomicterms). ThisleadstothetotalCoulomb
interaction
H =H +H +H +H +H . (1)
total S FS LS HFS vdW
e2 1 1
V = +
sHterurec,tuHrSe iHsathmeilStcohnria¨ond,inwgehricHhamcainltobneianap,pHrFoSxiimsathteedfinaes C 4πǫ0 |R~A−R~B| |~xA−~xB|
(see Chap. 34 of Ref. [23]) 1 1
. (5)
− ~xA R~B − ~xB R~A !
~p4 1 ~2g L~ S~ | − | | − |
H = i + α s i· i
FS i=A,B"−8m3c2 2 (cid:18)2m2c(cid:19) |~ri|3 One then uses the fact that the separation |R~A −R~B|
X betweenthetwonuclei(protons)ismuchlargerthanthat
~3
+ 4παδ(3)(~r ) , (2) between a given proton and its respective electron, that
8m2c i is, much larger than both ~r = ~x R~ and ~r =
(cid:21) A A A B
| | | − | | |
3
~x R~ . One then writes~x R~ =~r +(R~ R~ ) where d~ = e~r is the dipole operator for atom i (for
B B A B A A B i i
| − | − −
and ~x R~ =~r +(R~ R~ ). Expanding in ~r and atoms with more than one electron, one has to sum over
B A B B A A
~r , one−obtains − all the electrons in the atoms i = A,B). The R~ and
B A
R~ are the positions of the atomic nuclei, and E~ de-
B
e2 ~rA ~rB 3(~rA Rˆ)(~rB Rˆ) notes the operator of the quantized electric field. An
H = · − · ·
vdW 4πǫ R3 elegant way of deriving the retarded Casimir–Polder in-
0
e2 teraction, described in Eq. (85.4) of Ref. [23], then con-
= 4πǫ R3 δkℓ−3RˆkRˆℓ rAkrBℓ, (6) sistsinthematchingofthescatteringamplitudeobtained
0 (cid:16) (cid:17) from quantum electrodynamics, against the effective in-
where R~ = R~ R~ , R = R~ and Rˆ = R~/R. The teratomic interaction Hamiltonian. Alternative deriva-
A B
− | | tions use time-ordered perturbation theory [28].
indices k and ℓ corresponding to the Cartesian coordi-
The functional form of the interaction depends on the
natesaresummedover(Einsteinsummationconvention).
distance range In the van der Waals range (9) of inter-
The van der Waals interaction term, for a 2S–1S sys-
atomic distances, the interaction of ground-state atoms
tem, has vanishing elements in first-order perturbation
isoftheusualR−6 functionalform. Thisremainsvalidif
theory. Both atoms A and B have to undergo a virtual
oneatomisinametastableexcitedstate. Intheso-called
dipole transition to a P state for a nonvanishing effect,
Casimir–Polderrange,
and the leading-order van der Waals interaction is ob-
tained in second-order perturbation theory, leading to a ~
1/R6 interaction energy. The propagator denominator R≫ α2mc, (11)
in the standard Rayleigh–Schr¨odingerexpression for the
the interatomic distance is much larger than the wave-
second-order energy shift due to H is equal to the
vdW length of an optical transition, and the interaction of
sum of the virtual excitation energies of both atoms [4].
ground-state atoms has an R−7 function form. For
The close-range asymptotics of the interatomic interac-
the long-range interaction involving excited metastable
tionenergythusgoesas1/R6[4]. Foraninteratomicdis-
atoms, however, we have to distinguish a third range of
tance of R 30a ...100a (one hundred atomic units),
the energy∼shift i0s on the o0rder of 10−8...10−12 atomic very large interatomic distances,
units (Hartrees). The hierarchy ~c
Casimir–Polder II (or Lamb shift): R , (12)
≫
H H H H (7) L
h vdWi≪h HFSi≪h LSi≪h FSi which we would like to refer to as the Lamb shift range.
Here, is the Lamb shift energy. For metastable atoms,
isthusfulfilled forR&30a0. Forsufficientlylargeinter- the CaLsimir–Polder range is bounded from above by the
atomicdistance,theDiracδ potentialoftheHFSactsas
Lambshiftrange,andthecondition(11)shouldbemod-
a perturbation and can be treated as such, and we shall
ified to read
focus on this regime in the current manuscript.
For clarity, we should point out that the Hamilto- ~c ~
Casimir–PolderI: R , (13)
nian (6) remains valid in the nonretardation approxima- ≫ ≫ α2mc
L
tion. One can understand retardation as follows: When Forthe1S–1Sinteraction,theinteractionenergyreaches
the phase of the atomic oscillationduring a virtualtran- the Casimir–Polder asymptotic form, proportional to
sitionchangesappreciablyonthetimescaleittakeslight 1/R7, in both regimes described by Eqs. (12) and (13).
to travelthe interatomicseparationdistance R, thenthe For the 2S–1S interaction, it is only in the very long-
retardedformof the vander Waals interactionhas to be range regime (12) that we have a R−7 interaction,
used. Thecriterionforthe validityofthenonretardation with competing oscillatoryterms [29–31]proportionalto
approximation thus is ( 4/R2)cos[ R/(~c)].
L L
A further complication arises. The state with
R ~ a
= 0 , (8) atom A in an excited state and atom B in the
c ≪ Eh αc ground state, 2S A 1S B, is degenerate with the state
| i | i
1S 2S with the quantum numbers reversed among
or, more precisely, | iA| iB
theatoms. Thereisnodirectfirst-ordercouplingbetween
a0 = αm~c ≪R≪ α2~mc = aα0 , (9) i|n2SteirAac|1tiSoinB(6a)n,dbu|1tSiniAse|2cSoniBdodrudeert,oanthoeff-vdainagdoenraWltearamls
is obtained which is of the same order-of-magnitude as
if we take into account that substantial overlap of the the diagonal term, i.e., the term with the same in and
electronic wavefunctions is to be avoided. out states. The Hamiltonian matrix in the basis of the
The retarded interatomic interaction cannot be ob- degenerate states 2S 1S and 1S 2S has off-
A B A B
| i | i | i | i
tained on the basis of Eq. (6) alone; one has to use the diagonal(exchange)termsofsecondorderinthevander
atom-field interaction term [see Eq. (85.4) of Ref. [23]], Waals interaction[3]. The energy eigenvalues and eigen-
states are easily found in the degenerate basis and are
V(t)= E~(R~ ,t) d~ (t) E~(R~ ,t) d~ (t), (10) studied here in Sec. IV.
A A B B
− · − ·
4
III. 2S–1S DIRECT INTERACTION point of the investigations (see, e.g., Ref. [22]),
∞
~
A. Formalism E(dir)W(R)= dωα (iω)α (iω)
A;B − πc4(4πǫ )2 A B
0 Z
0
According to Eq. (85.17) in Chap. 85 of Ref. [23], the ω4 c c 2
e−2ωR/c 1+2 +5
interactionenergy between two atoms A andB in states × R2 ωR ωR
A and B is given by (cid:20) (cid:16) (cid:17) (cid:16) (cid:17)
| i | i c 3 c 4
+6 +3 . (16)
ωR ωR
(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
∞
E(dir)(R)=Re i~ dωα (ω)α (ω)e2iωR/c ω4 Wedonotexplicitlyindicatethe“realpart”ontheright-
A;B πc4(4πǫ )2 A B R2 handsideoftheaboveequation,becausethe polarizabil-
0 Z
0 ity α (iω) is manifestly real if we set ǫ = 0 in Eq. (15),
A
c c 2 c 3 c 4 and there are no poles near the integration contour in
1+2i 5 6i +3 .
× ωR − ωR − ωR ωR Eq. (16) to be considered. If both atoms are in their
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) (14) 1S groundstate,thentheexpressions(14)and(16)are
| i
equal [E(dir) (R) = E(dir)W(R)], and the Wick rotation
1S;1S 1S;1S
is permissible.
Herethesuperscript(dir)standsfor“direct”,asweantic- Let us now study the case A = 2S and B = 1S
| i | i | i | i
ipate that this interaction energy is to be supplemented foratomichydrogenasaparadigmaticexampleofalong-
by the so-called exchange interaction, to be discussed in range interaction involving a metastable excited state.
Sec. IV. The integral (14) constitutes the generalization In this case, the Wick rotated integral (16) is not equal
of the second-order van der Waals shift given by the ap- to (14), and extra care is needed [see also App. A]. The
plicationof Eq.(6), to the long-rangelimit, where retar- dipole polarizability α can naturally be split into two
2S
dation sets in. Eq. (14) contains the atom-field interac- contributions, the first of which is due to the quasi-
tion at the lowest relevant order in the elastic scattering degenerate 2P and 2P states which are dis-
1/2 3/2
case, where the initial and final states are identical (e.g. placed from 2S only by the Lamb shift and by the fine
(cid:12)| i (cid:11) (cid:12) (cid:11)
all photons emitted are reabsorbed and vice versa). We structure, re(cid:12)spectively. T(cid:12)he second contribution is due
here restrict the discussion to the leading effect in the to nP states with principal quantum number n 3. Af-
≥
multipole expansion, given by the dipole polarizability terdoingtheangularalgebraforthe 2P and 2P
1/2 3/2
α (i=A,B). Thedesignationoftherealpartofthe en- states whose oscillator strengths [32] with respect to 2S
i
ergy shift is necessary because the integrand constitutes are distributed in a ratio 1 : 2, we ob(cid:12)(cid:12)tain (cid:11) (cid:12)(cid:12) (cid:11)
3 3
acomplexratherthanrealquantity,andthe poles ofthe
integrand are displaced from the real axis according to
α (ω)=α (ω)+α (ω), (17a)
2S 2S 2S
the Feynman prescription. For the dipole polarizability
αA(ω) (of atom A), we have α2S(ω)=P2S(ω)+P2S(−ω), (17b)
e
α (ω)=P (ω)+P ( ω), (17c)
2S 2S 2S
−
e2 2S ~r 2P(m=µ) 2
αA(ω)=PA(ω)+PA(−ω), Pe2S(ω)= e9 |h e| |+~ω iǫ i| (17d)
e2 1 Xµ −L −
P (ω)= ψ ~r ~r ψ
A 3 A H E +~ω iǫ A 2e2 2S ~r 2P(m=µ) 2
(cid:28) (cid:12) − A − (cid:12) (cid:29) + |h | | i|
= e2 3(cid:12)(cid:12)(cid:12) |hψA|~r|ψni|2 ,(cid:12)(cid:12)(cid:12) (15) 9 Xµ F +~ω−iǫ
3 E E +~ω iǫ 1 2
n i=1 n− A − =3e2a2 + ,
XX 0 +~ω iǫ +~ω iǫ
(cid:18)−L − F − (cid:19)
e2 2S ~r nP(m=µ) 2
P (ω)= |h | | i| . (17e)
where H in the propagator denominator denotes the 2S 3 E E +~ω iǫ
n 2S
Schr¨odinger Hamiltonian of the relevant atom. The ǫ nX≥3Xµ − −
parameter in Eq. (15), ensures that the integration (14) e
is carried along the Feynman contour; the limit ǫ 0+ The nondegenerate contribution to the 2S polarizability
→
is taken after the integration is carried out. Under ap- is denoted as α (the quasi-degenerate 2P levels are
2S
propriateconditions,whicharediscussedindetailbelow, excluded). The quasi-degenerate2P levels are contained
we may perform a Wick rotation dω idω in the inte- inα . AllsumsareoverthenonrelativisticP stateswith
2S e
→
gral (14). The resulting Wick (W) rotated expression is magneticprojectionquantumnumbersµ= 1,0,1. The
−
the familiar one which is usually taken as the starting Lambshift energy and the fine structure energy are
L F
5
defined as which is valid for a and b real (regardless of their sign).
A change in integration limits to the interval (0, ) can
E(2S1/2)−E(2P1/2)≡L, be absorbed in a prefactor 2. The result for D6(∞2S;1S)
E(2P ) E(2S ) . (18) reads
3/2 1/2
− ≡F
3 ~ 2e2
The leading-order expressions for and read as = D (2S;1S)= 1S ~r k 2
α α4mc2 ln[α−2] and = α4mc2L/32, reFspectivelyL[26] 6 π(4πǫ0)2 3 |h | | i|
6π F Xk
[see also Eq. (3)]. 2e2
2S ~r 2P(m=µ) 2
× 9 |h | | i|
µ
X
B. van der Waals range a0 ≪R≪a0/α π 1 2
+
× 2~ E E E E +
We investigate the 2S–1S interaction in the van der (cid:18) k− 1S −L k− 1S F(cid:19)
Waals regime (9). There is no exponential or oscillatory 3 ~ 2e2 1S ~r k 2
|h | | i|
suppression of any atomic transition in this regime, but ≈ π(4πǫ0)2 ( 3 Ek E1S )
k −
we can approximate X
2e2 π
27a2 3
EA(d;iBr)(R)=Reπc4(i4~πǫ0)2 Z0∞dωαA(ω)αB(ω)e2iωR/c ≈×π3(cid:18)(~4πe92ǫa)20(cid:0)2 {α01(cid:1)S(cid:19)(0×)} 2×~(6)×32π~
0
ω4 c c 2 c 3 c 4 243
1+2i 5 6i +3 = E a6, (24)
× R2 ωR − ωR − ωR ωR 2 h 0
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
∞
3 ~ where we took the limit 0, 0 at the end of the
≈ π(4πǫ )2R6 Rei dωαA(ω)αB(ω). (19) calculation. We have useLd→the knFow→n result
0 Z
0 9e2a2
α (0)= 0 , (25)
The functional form therefore is of the van der Waals 1S 2 E
h
type
where E = α2mc2 is the Hartree energy. We can now
h
E(dir)(R) D6(A;B), (20) give a more thorough analysis of the discrepancy of the
A;B ≈− R6 resultsforthe(2S;1S)vanderWaalscoefficientreported
inRefs.[3–5]. Namely,the denominatora+binEq.(23)
with the van der Waals coefficient
just corresponds to the sum of the excitation energies of
∞ the two atoms in the calculation of the van der Waals
3 ~
D (A;B)= Re i dωα (ω)α (ω) . coefficient; the contribution of a virtual P state in one
6 π(4πǫ0)2 − Z A B  of the atoms is seen to be nonvanishing even if it is
0
displaced from the reference state only by an infinites-
 (21)
imal shift a = , 0. By contrast, if one takes the
For the 2S–1S interaction, this implies that
L F →
limit , 0 too early, i.e., before evaluating the in-
3 ~ tegralL(2F3),→then in Eq. (17b), one obtains α (ω) = 0,
D (2S;1S)= 2S
6 π(4πǫ0)2 because the two terms P2S(±ω) just cancel each other.
Or, expressed more concisely, because of the exact ener-
∞
getic degeneracy of the 2S and 2P states in the nonrel-
Re i dω [α (ω)+α (ω)] α (ω) (22a)
× − 2S 2S 1S  ativistic theory, the virtual 2P states are excluded from
Z
0 the sum over virtual states in the nonrelativistic expres-
=D (2S;1S)+D (2S;1S)e.  (22b) sion of the polarizability, which leads to the erroneous
6 6
result reported in Refs. [3, 5]. Only if the formulation
For A = 2S and eB = 1S , D6 therefore is the sum of the nonrelativistic expression of the polarizability is
| i | i | i | i
oftwocontributionsD andD ,whichcorrespondtothe enhanced by the fine structure and Lamb shift denom-
6 6
degenerate α and nondegenerate α contributions to inators, as in Eq. (17), can we obtain the missing con-
2S 2S
the 2S polarizability, respecteively. The degenerate con- tribution D6(2S;1S) given in Eq. (24). The contribu-
tribution to D can be handled analytically. We use the tionofthe quasi-degeneratelevelsis more obviousin the
6 e
integral identity sum-over-states approach chosen in Ref. [4], where ac-
cordingto Eq.(23), the sumofthe excitationenergiesof
∞
i ab 1 both atoms enters the propagator denominator [see also
dx , Eqs. (12a) and (12b) of Ref. [4]].
− π−Z∞ (a−iǫ)2−x2 (b−iǫ)2−x2 ǫ→→0+ a+b For the nondegenerate contribution, we can perform
h i h i (23) the Wick rotation and obtain the following integral rep-
6
resentation A numerical integration of Eq. (26) then yields the
following value for D (2S;1S),
∞ 6
3 ~
D6(2S;1S)= π(4πǫ0)2Re−iZ dωα2S(ω)α1S(ω) D6(2S;1Se)=55.252266285Eha60. (28)
0
e 3 ~ ∞ e  We have veerified this result using discrete numerical
= π(4πǫ )2 dωα2S(iω)α1S(iω), methods [39], where the radial Schr¨odinger equation is
0 Z0 evaluated on a lattice, and a discrete pseudospectrum
(26)
(due to the finite size of the lattice) represents the con-
which is convenient for a numericeal evaluation. Namely,
tinuumspectrum. TheresultforD (2S;1S)accordingto
according to Eq. (15), one can write the corresponding 6
Table VI of Ref. [5] reads 56.7999E a6, while according
polarizabilitiesasthesumovertwomatrixelementsP(ω) h 0
toTable2ofRef.[3],itis(56.5 0.5)E a6. Bothresults
and P( ω) of a resolvent operator, where the P matrix ± h 0
− are not in perfect agreementwith our result, though nu-
elementscanbewrittenintermsofhypergeometricfunc-
merically close. This observation is consistent with the
tions. The calculation of a convenient representation of
derivations outlined in Refs. [3, 5], which suggest that
the polarizability of low-lying S states [33–35] becomes
the resultsreportedin the citedinvestigationmay corre-
easierifoneusesacoordinate-spaceintegrationbasedon
spond to the nondegenerate contribution. The total van
theSturmiandecompositionoftheradialhydrogenGreen
der Waals coefficients D is obtained as the sum of the
functionintermsofLaguerrepolynomials[36]. Afterthe 6
contributions given in Eqs. (24) and (28),
radialintegrals,oneevaluatesthesumovertheSturmian
integrals in terms of hypergeometric functions with the
D (2S;1S)=D (2S;1S)+D (2S;1S)
6 6 6
helpofformulascontainedinRef.[37]. Theresultofthis
calculation for the ground state is =176.752266285Eha60, (29)
e
e2a2 2t2 where we confirm all significant digits of the previously
P1S(ω)=− E 0 3(1 t)5(1+t)4 38t7+26t6 reported result [4] of 176.752. For the 1S–1S interac-
+19t5 19t4 h 12(cid:20)t3+−12t2+3t 3(cid:0) 6ti.o4n9,9w02e6co7n05firEmath6,eaknndowadndreasfuelwt[d4i,g4it0s]ooffnDu6m(1eSri;c1aSls)ig=-
− − − h 0
256t9 (cid:1) 1 t 2 nificance. In particular, this result shows that the result
+3(t−1)5(t+1)5 2F1 1,−t,1−t,(cid:18)1+−t(cid:19) !#, feoqruDal6(t1oSa;1rSa)tiiosnnaulmneurmicbalelry.clWoseetsoho123u,ldbuatdndottheaxtactthlye
2~ω −1/2 numerical accuracy of the strictly nonrelativistic results
t= 1+ , (27a) given in Eqs. (28) and (29) extends to all digits indi-
α2mc2
(cid:18) (cid:19) cated. However, reduced-mass, relativistic and radiative
corrections contribute on the level of 10−4...10−3. For
where
definiteness,weshouldalsoclarifythattheelectronmass
e2 1S ~r nP(m=µ) 2 m is used as the mass of the hydrogen atom, not the re-
P (ω)= |h | | i| , (27b)
1S 3 E E +~ω iǫ duced mass of the electron-proton system (see also the
n 1S
nX≥3Xµ − − discussion in Sec. VI).
and the sum includes the continuum. We here take the Analternativetreatmentispossibleinthepresentvan
opportunity to correct a typographical error in Eq. (3a) derWaalsrange. Thereexistsanintegralidentitysimilar
of Ref. [38] which led to an inconsistent sign of the term to (23), namely
involvingthe hypergeometricfunction. For the 2S state, 1 ∞ dx sgn(a′) sgn(b′)
one obtains the nondegenerate matrix element a′b′ = .
π (a′2+x2)(b′2+x2) a′ + b′
Z−∞ | | | |
e2a2 16τ2 (30)
P (ω)= 0 (1181τ8 314τ7
2S E 3(τ 1)6(1+τ)4 − Thetwointegrals(23)and(30)arethusequalfora+b=
h (cid:20) − a′+b′; if and only if a′ and b′ are both positive.
e 16τ6 166τ5+14τ4+138τ3 48τ2 42τ +21) Notice from (15) and (22a) that D (2S;1S) is given
− − − − 6
16384τ9(4τ2 1) 1 τ 2 by an integral of the type (23), namely, by
− F 1, 2τ,1 2τ, −
− 3(τ −1)6(τ +1)6 2 1 − − (cid:18)1+τ(cid:19) ! 4~ e4 ∞
D (2S;1S) Re i dω
72τ2 8~ω −1/2 6 ≡ − 3π(4πǫ )2
− 1 τ2 , τ = 1+ α2mc2 . (27c) 0 XmnZ0
− (cid:21) (cid:18) (cid:19) (Em E1S) 1S ~r m m ~r 1S
− h | | i·h | | i
Indeed,the2P stateisexcludedfromthesumoverstates × (E E iǫ)2 (~ω)2
inEq.(27c)bythesubtractionoftheterm72τ2/(τ2 1): m− 1S − −
one can verify that the expression (27c) is finite in−the (Eh E ) 2S ~r n n ~r 2Si
n 2S
− h | | i·h | | i. (31)
leinmeirtgyτω→ 10,. which is equivalent to vanishing photon × (En−E2S −iǫ)2−(~ω)2
→ h i
7
AtthispointwemaynotperformtheWickrotationthat We finally obtain
takes us from an integral of the type (23) to an integral
of the type (30). Indeed, for n = 2P1/2, we have b = 3 ~ ∞
E E = < 0 and the conditions for the equality D (2S;1S)= dωα (iω)α (iω)
n− 2S −L 6 π(4πǫ )2 1S(2S) 2S(1S)
of (23) and (30) is not fulfilled. However,as was noticed 0 Z
0
byDealandYoung in[4], anyintegralofthe type (30)is
=176.752266285E a6. (36)
equivalenttoanintegralofthetype(23)providedweare h 0
able to replacethe (possibly negative)quantities a andb
This matches the value (29) found by the previously fol-
bytwopositive quantitiesa′ andb′ sothata+b=a′+b′.
lowed method. With such a choice of the reference ener-
Hence, we can rewrite (31) as
gies in the denominators, we have shown that the Wick
∞ rotation is made automatically valid by the inequality
4~ e4 E > E for the virtual P states with energies E .
D6(2S;1S)=Re−i3π(4πǫ )2 dω Thmis pro1cSed2Sure also results in the automatic inclusionmof
0 mnZ
X0 the quasi-degenerate states.
E 1(E +E ) 1S ~r m m ~r 1S
m− 2 1S 2S h | | i·h | | i
× (cid:0) Em− 21(E1S +E(cid:1)2S)−iǫ 2−(~ω)2 C. Very large interatomic distance R≫~~~c/L
Eh(cid:0) 1(E +E ) 2S ~r n(cid:1) n ~r 2Si
n− 2 1S 2S h | | i·h | | i.
× (cid:0) En− 21(E1S +E(cid:1)2S)−iǫ 2−(~ω)2 For very large interatomic separations, the classic re-
sult is that of Casimir and Polder [41], and it is given,
h(cid:0) (cid:1) i (32)
when both atoms are in the ground state, by
In what follows we will make use of the space-savingno-
23 ~c 1
tation E(dir) (R) α (0)α (0). (37)
1S;1S ≈ −4π(4πǫ )2R7 1S 1S
0
1
E (E +E ). (33)
1S2S ≡ 2 1S 2S which can be obtained by the Wick-rotated version (16)
ofthe integral. When one ofthe atomssits in anexcited
Notice that for all single-atom hydrogen eigenstates [ex-
state,however(here,the2S state),thereisanextraterm
ceptfor1S,whichneverentersasavirtualstateintheex-
coming from the contribution of the pole that is picked
pressionof2S polarizabilities],wehaveE ,E >E .
m n 1S2S upwhencarryingouttheWickrotation. Thepolecorre-
In other words, identifying (32) with the model integral
sponds to the 2P level. We thus have two competing
(23), we have a and b positive. Hence the condition for 1/2
contributionsinthe very-long-rangelimit, thefirstbeing
theequalityof(23)and(30)isfulfilled. Wethenperform
the generalizationof Eq. (37) to the 2S–1S interaction,
the Wick rotation and rewrite (32) as
D6(2S;1S)= 34π~(4πeǫ4 )2 ∞dω E2(dSi;r1)SI(R)≈ −42π3(4π~ǫc0)2R17 α1S(0)α2S(0), (38)
0 mnZ
X0 the other being an oscillatory term [29–31] of the func-
E 1(E +E ) 1S ~r m m ~r 1S
m− 2 1S 2S h | | i·h | | i tional form
× (cid:0) Em− 21(E1S +(cid:1)E2S) 2+(~ω)2 e2 4 2 R
E h(cid:0)1(E +E ) 2S ~r(cid:1) n n ~ri2S E(dir)II(R) L cos L
× (cid:0) n−E2n−1S21(E1S2S+(cid:1)Eh2S)|2|+i(~·hω)|2 | i. 2S;1S ∼ (4πǫ0)22RS2~r(cid:18)2P~c((cid:19)m=µ)(cid:18)2α~1cS((cid:19)0). (39)
h(cid:0) (cid:1) i (34) × µ |h | | i|
X
Weintroducethefollowingpolarizabilities,whichhave
The term E(dir)I is the Wick-rotated term (16) in the
the mean energy E in the propagatordenominators, 2S;1S
1S2S long-range limit. The term E(dir)II is the pole contri-
2S;1S
bution from the 2P level, which lies lower than the
e2 1 1/2
2S level. Inthe vanderWaalsrange(9), boththe Wick-
α (ω)= 1S ~r ~r 1S
1S(2S) 3 h | H E ~ω | i rotatedandpolecontributiondecayas1/R6. However,in
± − 1S2S ±
X the present large separationregime (12), we see that the
=P (ω)+P ( ω) , (35a)
1S(2S) 1S(2S) − poletermexhibits along-rangetailproportionaltoR−2.
e2 1 Forthe2S–1S interaction,itistheratio( R)/(~c)that
α2S(1S)(ω)= 3 h2S|~r H E ~ω~r|2Si determineswhichoneofthesepowersyieldLsthedominant
± − 1S2S ±
X contribution. Hence,wehaveatheregimechangearound
=P2S(1S)(ω)+P2S(1S)( ω) . (35b) R = ~c/ , with long-range tails extending beyond such
−
L
8
separations. Parametrically, using α5mc2, and quantum number are concerned, we are in the Casimir–
~c/ a /α4, one obtains the followLing∼estimates, Polder regime where the result is given by an R−7 inter-
0
L∼
action [only the virtual 2P state gives rise to an os-
1/2
E(dir)I(R) Eh , (40a) cillatory tail, and this occurs–without any change in the
2S;1S ∼ α4(R/a )7 principal quantum number—only for the 2S–1S interac-
0
α16 cos(2α4R/a )E tion]. The 2S–1S interaction wouldtherefore be propor-
E(dir)II(R) 0 h . (40b) tional to R−7 if the 2S polarizability were restricted to
2S;1S ∼ (R/a )2
0 the termα . However,the frequencyrangecorrespond-
2S
Both of these estimates are relevant for R & ~c/ . The ingtotheintermediatedistancerange(13)issolowthat
transition region where EI (R) becomes comLmensu- the frequeency-dependent quasi-degenerate polarizability
2S;1S α in the integral (14) is not exponentially suppressed.
rate with EII (R) is thus reached for 2S
2S;1S We thus have
R ~c a0 , EI (R) EII (R) α24E . E(dir) (R) E(dir) (R)
∼ ∼ α4 2S;1S ∼ 2S;1S ∼ h 2S;1S ≈ 2S;1S
L (41) ∞
~
The frequency shift in this region is of the order of =3 Re i dωα (0)α (ω) .
10−36Hz,andthusfartoosmalltobeofanyrelevancefor (4πǫ0)2  Z 1S 2S 
0
experiments. Inview ofthe prefactor 4 inEq.(39), the  (44)
L
same conclusion is reached as recently found in Ref. [42]
for atom-surfaceinteractions: Namely, for long-rangein- The static ground-state polarizability α (0) is given in
1S
teractions involving the metastable 2S state, a poten- Eq. (25). Furthermore, on the scale of distances in the
tially interesting oscillating long-range is found, but its intermediate range,we may approximate the Lamb shift
numerical coefficient is too small to be of significance. and the fine structure energy by zero after doing the in-
Ourvery-long-rangeregimeisgivenby(12). Expressed tegrals. This yields
in units of the Hartree energy E , the physical values of
h
the Lamb shift and fine structure energies are ∞
2 π
lim Re i dω L = . (45a)
=1.61 10−7E , (42a) L→0  ( iǫ)2 (~ω)2 ~
L × h Z0 −L− −
=1.67 10−6E 10 . (42b)  
F × h ≈ L Duetothedifferentpolestructureunderthesignchange
from the Lamb shift as compared to the fine structure
The long-rangeapproximationis thus valid in the region
transition ( <0, but >0), it is nontrivial to check
−L F
~c a E that
R = 0 h =8.206 108a =0.0434m. (43)
0
≫ α × ∞
L L 2 π
According to Eq. (41), the oscillatory tail and the 1/R7 Fli→m0Re−i dω ( iǫ)2F (~ω)2= ~ , (45b)
Casimir–Polder term have comparable magnitude as we Z0 F − −
enterthevery-long-rangeregime(12),buttheoscillatory  
The result for the asymptotics in the intermediate range
tail given in Eq. (40b) could be assumed to dominate
thus reads as
for distances exceeding the Lamb shift transition wave-
length. Thisconsideration,though,shouldbetakenwith
(dir) D6(2S;1S) 243 a0 6
a grain of salt. Namely, in the long-range limit, one has E (R)= = E . (46)
2S;1S − R6 − 2 h R
to take into consideration the fact that the width of the
(cid:16) (cid:17)
2P1/2 state is of the same order-of-magnitude (α5mc2) The interaction is thus still of the R−6 form, as it is in
as the Lamb shift itself [32]. For R ~c/ , the oscilla- the vander Waalsrange,butthe coefficientis reducedin
≫ L
tory tails are thus exponentially suppressedaccordingto magnitude as compared to Eq. (29).
the factor exp[2i( +iΓ/2)R)/(~c)] exp( 2ΓR/(~c)), A few words on the precise formulation of the inter-
L ∼ −
where Γ is the natural energy width of the 2P1/2 state. mediate distance range are perhaps in order. Namely, in
Still,itisofAcademicinteresttonotethattheoscillatory principle, one might argue that the intermediate range
long-range tail exists. shouldbeboundedfromaboveby~c/ ,insteadof~c/ ,
F L
as the former quantity is smaller than the latter. In
the rather narrow window where ~c/ < R < ~c/ ,
D. Intermediate distance a0/α≪R≪~~~c/L transitions between 2S and 2P statFes are suppresseLd
3/2
while those between 2S and 2P states are not. We
1/2
It is very interesting indeed to also investigate the in- do not dwell further on the details of this regime, be-
termediate rangeofinteratomicdistances,givenby (13). cause an order-of-magnitude estimate of the frequency
Thetreatmentbecomesalittlesophisticated. Namely,as shifts, analogous to the one carried out in Sec. IIIC, re-
far as virtual transitions with a change in the principal veals that they do not exceed 10−21Hz in the discussed
9
distance range. Mathematically speaking, the inequality where we define the two coefficients
R≪~c/LimpliesR≪~c/F becauseF andLareapart 2 e4 ′ 1S ~r m 2 2S ~r n 2
by only a single order-of-magnitude [see Eq. (42)]. The D (2S;1S)= h | | i| |h | | i ,
breygiamneu~mce/rFica<l cRalc<ul~atci/oLnc(saeneoSnelcy. VbeI)a.ccessed reliably 6 3(4πǫ0)2Xmn (cid:12)(cid:12)Em+En−(E1S +E2(S(cid:12)(cid:12)5)4a)
2 e4 ′ 1
M (2S;1S)=
IV. 2S–1S EXCHANGE INTERACTION 6 3(4πǫ0)2mn Em+En−(E1S +E2S)
X
1S ~r n n ~r 2S 2S ~r m m ~r 1S .
A. Formalism ×h | | i·h | | i h | | i·h | | i (54b)
It can be shown that D (2S;1S), as defined by (54a),
Wenowconsiderthe2S–1S exchangeinteraction. The 6
agreeswiththeearlierexpression(21). Theeigenenergies
states 1S 2S and 2S 1S are energetically de-
| iA| iB | iA| iB and corresponding eigenvectors of matrix (53) are
generate, which induces the need for special care in the
treatment of the van der Waals interaction. The general D M
6 6
eigenvalue problem reads as follows, E± =E0− R±6 , (55a)
1
(H +H ) Ψ =E Ψ , (47)
S vdW ψ = (Ψ Ψ ) , (55b)
| i | i ± 1 2
| i √2 | i±| i
whereH istheSchr¨odingerHamiltonian(sumoverboth
S
atoms). Inwhatfollowsweshallattempttogiveasome- so that we obtain a symmetry-dependent van der Waals
whatstreamlinedderivationofthevanderWaalsmixing coefficient
termresultingfromtheenergeticdegeneracy,whichcon-
C =D M (56)
firms the results obtained in Ref. [3]. The basis states 6 6± 6
are which is obtained from a direct and a mixing term, de-
pending on the sign in the coherent superposition (55b).
Ψ = 1S 2S , (48a)
| 1i | iA| iB Using the integralrepresentation(23), one canbring M
6
Ψ = 2S 1S . (48b)
| 2i | iA| iB into the form (31)
Thefirst-orderperturbationsto thesewavefunctionsare
∞
′ 4~ e4
1 M (2S;1S) Re i dω
|δΨj=1,2i=(cid:18)E0−HS(cid:19) HvdW|Ψj=1,2i, (49) 6 ≡ − 3π(4πǫ0)2 XmnZ0
where E0 = E1S + E2S is the unperturbed energy of (Em−E1S) h1S|~r|mi·hm|~r|2Si
the metastable, noninteracting two-atom system. The × (E E iǫ)2 (~ω)2
m 1S
prime on the Green function indicates that the degener- − − −
ate states have been excluded from the sum over virtual h(En E2S) 2S ~r n ni~r 1S
− h | | i·h | | i. (57)
states. One calculates the Hamiltonian matrix with ele- × (E E iǫ)2 (~ω)2
n 2S
ments − − −
h i
H =( Ψ + δΨ )(H +H )(Ψ + δΨ ) , Expressed in terms of polarizabilities, one obtains
ij i i S vdW j j
h | h | | i | i
(50) ∞
3i~
with i,j =1,2. The result has the structure
D (2S;1S)=Re − dωα (ω)α (ω) ,
6 π(4πǫ )2 1S 2S
E +X Y 0 Z
H = 0 , (51) 0
Y E0+X (58a)
(cid:18) (cid:19)
∞
where 3i~
X = ′ |h1S2S|HvdW|mni|2 , (52a) M6(2S;1S)=Reπ(−4πǫ0)2 Z dωα1S2S(ω)α∗1S2S(ω) ,
E +E (E +E ) 0
mn 1S 2S − m n (58b)
X
′ 2S1S H mn mn H 1S2S
Y = h | vdW| i h | vdW| i. where we define the mixed polarizabilities via
E +E (E +E )
1S 2S m n
Xmn − e2 1
(52b) α (ω)= A ~r ~r B (59a)
AB 3 h | H E iǫ ~ω | i
Again,theprimeonthesumdenotestheexclusionofthe ± − A− ±
X
reference state. This matrix thus assumes the form =P (ω)+P ( ω) , (59b)
AB AB
−
D6(2S;1S) M6(2S;1S) e2 1
E0− R6 − R6 αAB(ω)= 3 hA|~rH E iǫ ~ω~r|Bi (59c)
H =  , (53) ± − B − ±
M (2S;1S) D (2S;1S) X
6 E 6 =P (ω)+P ( ω) . (59d)
 − R6 0− R6  AB AB −
 
10
For the (2S;1S) system, one obtains B. van der Waals range a0≪R≪a0/α
InthevanderWaalsrange(9),weproceedinasimilar
way to Sec. IIIB, and have
P1S2S(ω)= eE2ah20729( 1+51ν22√)22(ν24+ν2)3 E2(mS;x1dS)(R)≈Re3πi(4π~ǫ )2R16 ∞dωα1S2S(ω)α1S2S(ω)
− − 0 Z
0
128 272ν2+120ν4+253ν6+972ν7+419ν8 ∞
× − 3i ~ 1
(cid:18) =Re dωα (ω)
1 ν 2 ν π (4πǫ )2R6 1S2S
1944ν7 F 1, ν;1 ν; − − , 0 Z
2 1 0
− − − 1+ν 2+ν
(cid:18) (cid:19)(cid:19) α (ω)+α (ω) . (62a)
2n2 ~ω −1/2 × 1S2S 1S2S
ν =neff 1+ α2emffc2 . (60) This can be rewr(cid:2)itten as (cid:3)
(cid:18) (cid:19) e
M (2S;1S)
E(mxd)(R)= 6 . (62b)
2S;1S − R6
where M (2S;1S) = M (2S;1S)+M (2S;1S) is the
Herewewilltypicallychoosetheeffectivequantumnum- 6 6 6
ber neff to be either 1 (which yields P1S2S) or 2 (which sum of the nondegenerate M6(2S;1S) and degenerate
yields P1S2S), as required for input into Eq. (58). An- M6(2S;1S) contributifons to the mixed van der Waals
other possibility, less physically transparent but quite coefficient, with notations ofbvious from (62a). As was
handy for numerical calculations, is to choose n such done before, we can, for the nondegeneratecontribution,
eff
that the reference energy E = α2mc2/ 2n2 in perform the Wick rotation. For the degenerate contri-
the propagator correspondsnteoff the−average (33) eofff the bution, we follow the same procedure as in Sec. IIIB,
energies of the n = 1 and n = 2 levels (see S(cid:0)ecs. I(cid:1)IIB centered on the integral identity (23). This yields
and IVB). The latter choice corresponds to n =
eff
917504 a 6
2 2/5. E(mxd)(R)= 18.630786871+ E 0
2S;1S − − 19683 h R
p (cid:18) a 6 (cid:19) (cid:16) (cid:17)
Taking retardation into account, the generalization = 27.983245543E 0 (63)
h
− R
of Eq. (55) (minus the unperturbed energy E ) to the
0 (cid:16) (cid:17)
Casimir-Polder energy is where we make use of (60), whence
M =27.983245543E a6, (64)
6 h 0
to be compared to D as given by Eq. (36). The
6
∞
i ~ ω4 two terms in Eq. (63) correspond to the nondegener-
E± =Reπc4(4πǫ )2 dωe2iωR/c R2 ate (−18.630786...) and degenerate (91197658034) contribu-
0 Z tions, respectively. Their sum matches the results found
0
in Refs. [3, 4].
α (ω) α (ω) α (ω) α∗ (ω)
× 1S 2S ± 1S2S 1S2S As was the case for the direct interaction (see
h c c 2 c 3 i c 4 Sec. IIIB), an alternative treatment exists whereby we
1+2i 5 6i +3 make use of the integral identities (23) and (30). This
× ωR − ωR − ωR ωR
(cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21) yields the following expressionfor the van der Waals co-
=E(dir) (R) E(mxd)(R) . (61) efficient M :
2S;1S ± 2S;1S 6
3 ~ ∞
M (2S;1S)= dω α (ω)2 (65)
6 π(4πǫ )2 | 1S2S |
0 Z0
This result generalizes Eq. (55a) to the Casimir–Polder where the mixed polarizability α1S2S with averagerefer-
regime. It involves the mixed polarizabilities defined in ence energy (33) is defined by
Eq.(59). WerefertothesecondsummandE(mxd) asthe
2S;1S e2 1
exchange term. Diagrammatically, it is obtained from a α (ω)= 1S ~r ~r 2S
1S2S 3 h | H E ~ω | i
process in which an initial |1SiA|2SiB atoms makes a X± − 1S2S ±
ptrhaontsointiso.nAtsowaa|s2SthieAc|a1sSeifBorsttahteedviriaectth2eSe–x1cShainntgeeraocfttiwono =P1S2S(ω)+P1S2S(−ω) . (66)
term, we can single out three different distance regimes A numerical calculation based on Eq. (65) confirms the
for the exchange term, which we now investigate. result given in Eq. (63).