The van der Waals interaction
Barry R. Holsteina)
Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003 and Institute
for Nuclear Theory, Department of Physics, University of Washington, Seattle, Washington 98195
共Received 14 August 2000; accepted 17 October 2000兲
The interaction between two neutral but polarizable systems at separation R, usually called the van
der Waals force, is discussed from different points of view. The change in character from 1/R 6 to
1/R 7 due to retardation is explained. © 2001 American Association of Physics Teachers.
关DOI: 10.1119/1.1341251兴
I. INTRODUCTION
H0 ⫽
The interaction between charged particles via the Coulomb interaction is one of the most important features in
physics and is familiar to any student of the subject. The way
in which electrons and protons bind to form the hydrogen
atom is also well known and is a staple of any quantum
mechanics course.1 However, less familiar is the interaction
between such bound systems at separation R, which is the
so-called van der Waals force and is of a completely different character from its Coulombic analog.2 That this must be
the case is clear from the fact that the hydrogen atom is
neutral, so that to lowest order there is no interaction. On the
other hand the system is polarizable, and thus can interact
with the other polarizable system, leading to a short-ranged
attraction which varies as 1/R 6 , and this feature is discussed
by a number of quantum mechanical references.3,4 Somewhat
less well known is the fact that at larger distances the character of the interaction changes and varies as
1/R 7 —discussion of this feature can be found, e.g., in the
quantum field theory book by Itzykson and Zuber.5 It is clear
that the origin of this change is retardation, i.e., the finite
propagation time of signals connecting the two systems, but
the precise way in which this modification comes about is
not so easy to calculate and is not generally presented.
The nature of the van der Waals force is quite topical at
present due to the possible importance of such effects for the
interactions of small color dipoles such as charmonium or
bottomonium,6 so it is useful to examine the physics of this
effect. In the next section, then, we review the usual textbook
discussion leading to the London ⬃1/R 6 interaction.7 Then
in Sec. III, we show how retardation effects modify the character of the force and change its asymptotic dependence to
the Casimir–Polder form ⬃1/R 7 . 8 In a brief concluding section we summarize our findings and discuss the relevance to
modern particle and nuclear physics.
II. STANDARD VAN DER WAALS INTERACTION
The basic physics of the van der Waals force can be understood from a simple one-dimensional model of the atom
which consists of electrons bound by harmonic oscillator
forces to heavy protons at fixed separation R in addition to
Coulomb interactions between the four charges9
1 2 1
1 2 1
p 1 ⫹ m ␻ 20 x 21 ⫹
p ⫹ m ␻ 20 x 22 ,
2m
2
2m 2 2
冉
Assuming that the atomic separation is large compared to
the size of the atom (RⰇx 1 ,x 2 ), we can approximate
H1 ⬇⫺2
Am. J. Phys. 69 共4兲, April 2001
共2兲
H⫽
2
p⫹
2m
⫹
⫹
冉
冉
冊
2
p⫺
1
2e 2
2
m ␻ 20 ⫺
x
⫹
2
4 ␲ R 3 ⫹ 2m
冊
1
2e 2
m ␻ 20 ⫹
x2 ,
2
4␲R3 ⫺
共3兲
i.e., in terms of independent harmonic oscillators with shifted
frequencies
␻ ⫾⫽
冑
␻ 20 ⫿
⯝ ␻ 0⫿
2e 2
4 ␲ mR 3
e2
e4
⫺
⫹¯ .
4 ␲ m ␻ 0 R 3 32␲ 2 m 2 ␻ 30 R 6
共4兲
The van der Waals potential is simply the shift in the ground
state 共zero point兲 energy due to the Coulomb interaction and
is found to be
冉 冊
1
1
1
e4
V 共 R 兲 ⫽ ␻ ⫹ ⫹ ␻ ⫺ ⫺2 ␻ 0 ⯝⫺
. 共5兲
2
2
2
32␲ 2 m 2 ␻ 30 R 6
We can write this result in a more familiar form by noting
that when an external electric field is applied to this system,
the leading order Hamiltonian becomes
H⫽H0 共 x 1 ,x 2 兲 ⫹eE 0 x 1 ⫹eE 0 x 2 ⫽H0 共 z 1 ,z 2 兲 ⫺
z i ⫽x i ⫹eE 0 /m ␻ 20 ,
with
tric dipole moment
e 2 E 20
m ␻ 20
共6兲
and corresponds to an induced elec-
␦ H 2e 2 E 0
⫽
.
␦ E 0 m ␻ 20
共7兲
Defining the electric polarizability ␣ E in the conventional
fashion, via d⫽4 ␲ ␣ E E 0 , we find ␣ E ⫽2e 2 /4␲ m ␻ 20 so that
with 共see Fig. 1兲
441
e 2x 1x 2
4␲R3
and the system can be diagonalized in terms of coordinates
x ⫾ ⫽(x 1 ⫾x 2 )/&, yielding
d⫽⫺
H⫽H0 ⫹H1 ,
冊
共1兲
e2 1
1
1
1
H1 ⫽
⫺
⫺
.
⫹
4 ␲ R R⫹x 1 ⫺x 2 R⫹x 1 R⫺x 2
http://ojps.aip.org/ajp/
© 2001 American Association of Physics Teachers
441
d 2 ⫽4 ␲ ␣ E E 1 共 R 兲 ⫽4 ␲ ␣ E
ex 1
.
4␲R3
共13兲
The electric field generated by this electric dipole moment
then acts back on the original atom, yielding an energy
⌬E vdW⬃⫺d 1 E 2 共 R 兲 ⫽⫺
Fig. 1. Simple one-dimensional model of interacting hydrogen atoms.
the van der Waals interaction can be written in the ‘‘London’’ form7
V 共 R 兲 ⫽⫺
␣ E2 ␻ 0
.
8R 6
共8兲
One can also derive Eq. 共8兲 via simple second-order perturbation theory
⌬E 2 ⫽V 共 R 兲 ⫽
兺
n⫽0
具 0 兩 H1 兩 n 典具 n 兩 H1 兩 0 典
E 0 ⫺E n
共9兲
.
具 l 兩 x 兩 0 典 ⫽ ␦ l,1
冑
1
,
2m ␻ 0
共10兲
Eq. 共9兲 becomes
冉 冊兺
冉 冊 冉冑 冊
2e 2
V共 R 兲⫽
4␲R3
2
2e 2
⫽
4␲R3
2
␦ n 1 ,1␦ n 2 ,1兩 具 1,1兩 x 1 x 2 兩 0,0典 兩 2
n 1 ,n 2
1
2m ␻ 0
共 n 1 ⫹n 2 兲 ␻ 0
4
⫺1
e4
⫽⫺
2 2 3 6,
2␻0
32␲ m ␻ 0 R
共11兲
in agreement with Eq. 共5兲.
It is useful to spend a bit of time examining the ‘‘physics’’
of this result. The form of the interaction potential, Eq. 共2兲,
can be understood in terms of the energy of the dipole moment of ‘‘atom’’ #2 (d 2 ⫽⫺ex 2 ) in the electric field created
by the dipole moment of ‘‘atom’’ #1,
具 0 兩 eE 0 x 1 兩 n 典具 n 兩 eE 0 x 1 兩 0 典
E 0 ⫺E n
1
⬅⫺ 4 ␲ ␣ E E 20 .
2
共15兲
We find then ␣ E ⬃e 2 具 x 21 典 / ␻ 0 and
␣ E2 ␻ 0
4␲R6
共16兲
so that it is this self-interaction energy which is responsible
for the London form—cf. Eq. 共8兲.
With this background in hand it is straightforward to move
to the physical 共three-dimensional兲 situation.11 In this case
the dipole moment generated by atom #1 (d1 ⫽er1 ) generates an electromagnetic potential
U 共 R兲 ⫽
d1 "R
,
4␲R3
共17兲
which means that at location R one has the electric field
E共 R兲 ⫽⫺ⵜ RU⫽⫺
e
关 r ⫺3R̂R̂"r1 兴 .
4␲R3 1
共18兲
The corresponding dipole–dipole interaction energy is
e2
关 r "r ⫺3r1 "R̂r2 "R̂ 兴 .
4␲R3 1 2
共19兲
U vdW⫽
e2
关 x x ⫹y 1 y 2 ⫺2z 1 z 2 兴 .
4␲R3 1 2
共20兲
The lowest order energy shift involving a pair of hydrogen
atoms is then
However, there is a shift at second order since at any given
instant of time there exists an instantaneous dipole moment
in, say, atom #1. The corresponding electric field at the position of atom #2 generates a correlated electric dipole moment due to its electric polarizability,
442
共14兲
,
Choosing the z axis along the direction R̂, we can write
⌬E 1 ⫽ 具 ␺ 0 兩 H1 兩 ␺ 0 典 ⫽0.
冉 冊
兺
n⫽0
共12兲
Of course, 具 x 1 典 ⫽ 具 x 2 典 ⫽0, i.e., there exists no average dipole
moment, so this energy change vanishes in first-order perturbation theory
e2
V 共 R 兲 ⫽⌬E 2 ⫽
4␲R3
⌬E 共 2 兲 ⫽
U vdW⫽⫺d2 "E共 R兲 ⫽
⫺ex 1
e 2x 1x 2
⫽⫺
.
H1 ⬃⫺d 2 E 2 共 d 1 兲 ⫽ex 2
4␲R3
4␲R3
4␲R6
which is the van der Waals interaction. What makes this
work, then, is the point that one can use the instantaneous
position of one atom to provide an action at a distance correlation with a second atom in the vicinity.
Finally, we note that the electric polarizability itself can be
extracted by calculating the shift in energy of the atom in the
presence of an external electric field E 0 in second-order perturbation theory10
⌬E vdW⬃
Since for a simple one-dimensional harmonic oscillator
e 2 x 21 ␣ E
2
兺
1兲
2兲
1兲
2兲
␺ 共1,0,0
兩 U vdW兩 ␺ 共1,0,0
␺ 共1,0,0
⌬E 1 ⫽ 具 ␺ 共1,0,0
典 ⫽0
and vanishes since 具 r1 典 ⫽ 具 r2 典 ⫽0. However, to second order
there exists a nonvanishing energy—this is the van der
Waals interaction
共2兲
1兲
1兲
2兲
兩 具 ␺ 共n,l,m
␺ n ⬘ l ⬘ m ⬘ 兩 x 1 x 2 ⫹y 1 y 2 ⫺2z 1 z 2 兩 ␺ 共1,0,0
␺ 共1,0,0
典兩2
nlm;n ⬘ l ⬘ ,m ⬘
Am. J. Phys., Vol. 69, No. 4, April 2001
共21兲
2E 10⫺E nl ⫺E n ⬘ l ⬘
共22兲
.
Barry R. Holstein
442
Since the state ␺ 1,0,0 is the ground state, the denominator is
always negative and it is thus clear that this result is nonzero,
although no exact evaluation is possible. Nevertheless, we
can obtain an approximate form by noting that selection rules
allow only electric dipole 共⌬ j⫽0,⫾1, parity change兲 excitation of both atoms so that, using closure, we can write
V共 R 兲⯝
冉 冊
e2
4␲R3
⫻
2
兺
nlm;n l m
⬘⬘ ⬘
冉 冊
e2
4␲R3
2
共2兲
Using
dius, we find
共23兲
where a 0 ⫽1/m ␣ is the Bohr ra-
6 ␣ a 50
6 .
R
共24兲
Expressing this result in terms of the electric polarizability
via
␣ E ⫽2 ␣ 兺
nlm
兩 具 ␺ nlm 兩 z 兩 ␺ 1,0,0 典 兩 2
⯝a 30 ,
E 10⫺E nl
e2
␣ a 60 ␻ 0 ␣ E2
⫽
,
a0 R6
R6
H⫽
共26兲
which is the London form.
III. RETARDATION
The discussion given in the previous section is standard,
but it is clear that it must be altered at large distances since
we have neglected the finite propagation time of electromagnetic interactions. Indeed, since the van der Waals force is
due to the self-interaction of the dipole moment of an atom
with the electric field generated by the correlated dipole moment of an atom at distance R, this simple nonrelativistic
action at a distance analysis must begin to break down when
the time required for a signal to travel from atom to atom
(⌬T 1 ⬃R) becomes comparable to or greater than a characteristic time 关 ⌬T 2 ⬃1/(E 10⫺E 21) 兴 associated with the evolution of the atom, i.e., when ␻ 0 R⭓1. For hydrogen this
transition occurs at R⬃បc/10 eV⬃a 0 / ␣ ⬃200 A. For distances comparable to or greater than this value, we must take
retardation into account. There are at least two ways in
which this can be accomplished, and we shall consider each
in turn.
A. Fourth-order perturbation theory
Perhaps the most straightforward way to understand the
effects of retardation is through a simple perturbation theory
evaluation. Of course, since the van der Waals interaction is
O(e 4 ) this must be a fourth-order perturbative calculation.
This is how the original calculation of Casimir and Polder
Am. J. Phys., Vol. 69, No. 4, April 2001
册
共27兲
which arises from expansion of the instantaneous Coulomb
interactions between the charged particles making up the
system兴. It is thus advantageous, as pointed out by Power
and Zienau,13 to isolate the important physics by working in
the gauge ␾ ⫽0 and using the electric dipole approximation,
wherein the variation of the vector potential over the atomic
size is neglected. In this case the Hamiltonian becomes
兺
j⫽1,2
V共 R 兲⬃
冋
e
e2 2
pi "A共 ri 兲 ⫹
A 共 ri 兲
m
2m
r1 "r2 ⫺3r1 "R̂r2 "R̂
,
4␲R3
共25兲
we have
443
兺
j⫽1,2
⫺
r1 "r2 ⫺3r1 "R̂r2 "R̂
4␲R3
leads to a rather complicated analysis involving thirteen
diagrams—including seagull terms 关i.e., the diagrams arising
from the term A2 (ri ) in Eq. 共27兲 involving a pair of photons
emitted from the same point, which diagramatically look like
a child’s drawing of a flying seagull兴 and instantaneous Coulomb interactions 关i.e., the term involving
1
具 ␺共1兲 ␺共2兲 兩x 2x 2
2E 10⫺2E 21 1,0,0 1,0,0 1 2
具 x 2i 典 ⫽ 具 r2 典 /3⫽a 20 ,
兺 H 共j 0 兲⫹H rad⫹e 2
j⫽1,2
⫹
1兲
兩 具 ␺ 共n,l,m
␺ n ⬘l ⬘m ⬘兩 x 1 x 2
1兲
2兲
⫹y 21 y 22 ⫹4z 21 z 22 兩 ␺ 共1,0,0
␺ 共1,0,0
典.
V 共 R 兲 ⯝⫺
H⫽
1
2E 10⫺2E 21
1兲
2兲
⫹y 1 y 2 ⫺2z 1 z 2 兩 ␺ 共1,0,0
␺ 共1,0,0
典兩2
⫽
was done.8 However, use of the usual Coulomb gauge
Hamiltonian for two neutral hydrogen atoms at distance R
interacting with the electromagnetic field12
H 共j 0 兲 ⫹H rad⫺
兺
j⫽1,2
eri "Et ,
共28兲
where Et represents the transverse electric field so that only
radiation photons are involved. The resulting fourth-order
diagrams which contribute to the van der Waals interaction
are now only six in number. They are shown in Fig. 2 and
can be divided into three classes. One class is that where the
interaction with the electric field by atom #1 occurs entirely
previous to the interaction with the field by atom #2. For
pedagogical simplicity, we suppose that the atom has only a
single excited state 共兩1典兲 connected to the ground state 共兩0典兲
via the electric dipole operator er, and for didactic purposes
we shall explicitly demonstrate how to evaluate one of the
six diagrams—Fig. 2共a兲, which is shown in more detail in
Fig. 3. We use
⌬E 共04 兲 ⫽
兺
n,l,k⫽0
具 0 兩 V 兩 n 典具 n 兩 V 兩 l 典具 l 兩 V 兩 k 典具 k 兩 V 兩 0 典
共 E 0 ⫺E n 兲共 E 0 ⫺E l 兲共 E 0 ⫺E k 兲
,
共29兲
where we have dropped self-energy terms, which do not contribute. In the gauge ␾ ⫽0 we can write the interaction as
V⫽e
⳵A
兺 r1 " ⳵ t t ,
j⫽1,2
共30兲
where A is the quantized radiation field. For the diagram
shown we identify E n ⫺E 0 ⫽ ␻ 10⫹k 2 , E l ⫺E 0 ⫽k 1 ⫹k 2 , and
E k ⫺E 0 ⫽ ␻ 10⫹k 1 , where ␻ 10⫽E 1 ⫺E 0 is the excitation energy. Also,
具 k 兩 V 兩 0 典 ⫽iek 1 ⑀ˆ 1 • 具 1 兩 r 1 e ik1 • 共 r1 ⫹X1 兲 兩 0 典 ,
where X1 is the location of the atom, with similar expressions for the remaining three transition amplitudes. Putting
all these together we find
Barry R. Holstein
443
兺
k , ⑀ˆ
2
⬅
1
冕
d 3k 1
共 2 ␲ 兲 3 2k 1
.
兺
⑀ˆ
1
Now combine this result with that obtained by interchanging atoms #1, #2 and include the corresponding contribution
from Fig. 2共b兲. The result is
⌬E a⫹b ⫽⫺e 4
兺 兺
k1 , ⑀ˆ 1 k2 , ⑀ˆ 2
k 21 k 22 e i 共 k1 ⫹k2 兲 "R
⫻
冋
⫹
⑀ˆ 1* "具 0 兩 r2 兩 1 典 ⑀ˆ 2 "具 1 兩 r2 兩 0 典 ⑀ˆ *
ˆ 1 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r1 兩 1 典 ⑀
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 1 兲共 k 1 ⫹k 2 兲
⑀ˆ 2* "具 0 兩 r2 兩 1 典 ⑀ˆ 1 "具 1 兩 r2 兩 0 典 ⑀ˆ *
ˆ 1 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r1 兩 1 典 ⑀
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 2 兲共 k 1 ⫹k 2 兲
册
⫹#1↔#2 .
共32兲
Similarly one finds contributions from Figs. 2共c兲 and 2共d兲,
⌬E c⫹d ⫽⫺e 4
Fig. 2. Time-dependent perturbation theory diagrams contributing to the van
der Waals interaction. Here the dashed line indicates a photon exchange.
⌬E 2a ⫽⫺e 4
兺 兺
k1 , ⑀ˆ 1 k2 , ⑀ˆ 2
兺 兺
k1 , ⑀ˆ 1 k2 , ⑀ˆ 2
⫻
冋
⫹
⑀ˆ *
ˆ 1 "具 1 兩 r2 兩 0 典 ⑀ˆ 2* "具 0 兩 r1 兩 1 典 ⑀ˆ 1 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r2 兩 1 典 ⑀
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 2 兲 2 ␻ 10
⑀ˆ *
ˆ 1 "具 1 兩 r2 兩 0 典 ⑀ˆ *
ˆ 1 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r2 兩 1 典 ⑀
2 "具 0 兩 r1 兩 1 典 ⑀
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 2 兲 2 ␻ 10
⫹#1↔#2
k 21 k 22 e i 共 k1 ⫹k2 兲 "R
⑀ˆ 2* "具 0 兩 r2 兩 1 典 ⑀ˆ 1 "具 1 兩 r2 兩 0 典 ⑀ˆ *
ˆ 1 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r1 兩 1 典 ⑀
⫻
,
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 2 兲共 k 1 ⫹k 2 兲
共31兲
k 21 k 22 e i 共 k1 ⫹k2 兲 "R
册
共33兲
and from Figs. 2共e兲 and 2共f兲,
⌬E e⫹ f ⫽⫺e 4
兺 兺
k1 , ⑀ˆ 1 k2 , ⑀ˆ 2
k 21 k 22 e i 共 k1 ⫹k2 兲 "R
⫻
冋
⫹
⑀ˆ *
ˆ 1 "具 1 兩 r2 兩 0 典 ⑀ˆ 1* "具 0 兩 r1 兩 1 典 ⑀ˆ 2 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r2 兩 1 典 ⑀
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 2 兲共 2 ␻ 10⫹k 1 ⫹k 2 兲
where
⑀ˆ *
ˆ 1 "具 1 兩 r2 兩 0 典 ⑀ˆ 1* "具 0 兩 r1 兩 1 典 ⑀ˆ 2 "具 1 兩 r1 兩 0 典
2 "具 0 兩 r2 兩 1 典 ⑀
共 ␻ 10⫹k 1 兲共 ␻ 10⫹k 1 兲共 2 ␻ 10⫹k 1 ⫹k 2 兲
册
⫹#1↔#2 .
共34兲
Consider first the contribution from Figs. 2共a兲 and 2共b兲. Summing over polarization states, Figs. 2共a兲 and 2共b兲 become
⌬E a⫹b ⫽⫺e 4
⫻
⫻
Fig. 3. One of the time-dependent perturbation theory diagrams contributing
to the van der Waals interaction. Here the dashed line indicates a photon
exchange.
444
Am. J. Phys., Vol. 69, No. 4, April 2001
冕
d 3k 1
d 3k 2
k 2 k 2 e i 共 k1 ⫹k2 兲 "R
2k 1 共 2 ␲ 兲 3 2k 2 共 2 ␲ 兲 3 1 2
兩 具 1 兩 z 1 兩 0 典 兩 2 兩 具 1 兩 z 2 兩 0 典 兩 2 共 1⫹ 共 k̂ 1 "k̂ 2 兲 2 兲
共 ␻ 10⫹k 1 兲共 k 1 ⫹k 2 兲
冉
冊
1
1
⫹
.
␻ 10⫹k 2 ␻ 10⫹k 1
共35兲
Consider first the long distance limit ( ␻ 10RⰇ1) wherein retardation effects are important. Then since, due to the exponential, the sum over photon momenta extends only to k 1 ,
k 2 ⭐1/R, we can write
Barry R. Holstein
444
␻ 10RⰇ1
⌬E a⫹b → ⫺4
e4
2
␻ 10
冕
⫺23共 ␣ E2 ⫹ ␤ 2M 兲 ⫹7 共 ␣ E ␤ M ⫹ ␤ M ␣ E 兲
V共 R 兲⫽
,
4␲R7
d 3k 1
d 3k 2
3
2k 1 共 2 ␲ 兲 2k 2 共 2 ␲ 兲 3
⫻k 21 k 22 e i 共 k1 ⫹k2 兲 "R
⫻
where we have used the result
兩 具 1 兩 z 1 兩 0 典 兩 2 兩 具 1 兩 z 2 兩 0 典 兩 2 共 1⫹ 共 k̂ 1 "k̂ 2 兲 2 兲
.
k 1 ⫹k 2
共36兲
Using the definition of electric polarizability
␣ E⫽
兩具n兩z兩0典兩2 e2 兩具1兩z兩0典兩2
e2
⫽
,
2 ␲ n⫽0 E n ⫺E 0
2␲
␻ 10
兺
共37兲
this becomes
␻ 10RⰇ1
␣ E2
→ ⫺
共 2␲ 兲4
⌬E a⫹b
冕
⫻
d 3 k 1 d 3 k 2 k 1 k 2 e i 共 k1 ⫹k2 兲 "R
冉
1⫹ 共 k̂ 1 "k̂ 2 兲 2
k 1 ⫹k 2
␣ E2
⬅⫺
J .
共 2␲ 兲4 1
冊
共38兲
The integration can easily be performed, as shown in Appendix A, and the result is
⌬E a⫹b ⫽⫺
23␣ E2
4␲R7
共39兲
,
which is the result given by Casimir and Polder.8 Similarly,
the contributions from the remaining diagrams become, in
this limit,
冕
d 3 k 1 d 3 k 2 k 1 k 2 e i 共 k1 ⫹k2 兲 "R
␣ E2
→ ⫺
共 2 ␲ 兲 4 2 ␻ 10
␻ 10RⰆ1
⌬E c⫹d → ⫺
兺
d 3 k 1 d 3 k 2 k 1 k 2 e i 共 k1 ⫹k2 兲 "R共 1⫹ 共 k̂ 1 "k̂ 2 兲 2 兲 .
共 ⑀ˆ 1 "⑀ˆ 2 兲 2 ⫽1⫹ 共 k̂ 1 "k̂ 2 兲 2 ⫽
兺
共 ⑀ˆ 1 Ãk̂ 1 "⑀ˆ 2 兲 2 ⫽2k̂ 1 "k̂ 2 ⫽
兺
⑀ˆ 1 , ⑀ˆ 2
共41兲
兺
⑀ˆ 1 , ⑀ˆ 2
共 ⑀ˆ 1 "⑀ˆ 2 Ãk̂ 2 兲 2 .
共42兲
Then the overall van der Waals interaction at long distance
assumes the familiar form
445
冕
d 3 k 1 d 3 k 2 i 共 k ⫹k 兲 "R
e 1 2
共 2␲ 兲3 共 2␲ 兲3
共45兲
Am. J. Phys., Vol. 69, No. 4, April 2001
3 ␻ 10␣ E2
4R 6
共46兲
,
which is the London form. For this short distance situation
the contribution of the remaining diagrams is found to be
⌬E a⫹b ⫽⌬E e⫹ f
␻ 10RⰆ1
→ ⫺
2
␣ E2 ␻ 10
共 2␲ 兲4
⫻
共 ⑀ˆ Ãk̂ 1 "⑀ˆ 2 Ãk̂ 2 兲 2
the contribution to the van der Waals interaction from a pair
of magnetic interactions is found simply by replacing ␣ E2
→ ␤ 2M in Eq. 共39兲. Similarly, we can include interactions
involving electric and magnetic polarizabilities via the identity
⑀ˆ 1 , ⑀ˆ 2
e4
2 ␻ 10
⫻ 兩 具 1 兩 z 1 兩 0 典 兩 2 兩 具 1 兩 z 2 兩 0 典 兩 2 共 1⫹ 共 k̂ 1 "k̂ 2 兲 2 兲 .
⌬E c⫹d ⫽⫺
共40兲
By simple scaling arguments, it is easy to see that ⌬E c⫹d
⫽⌬E e⫹ f ⬃ ␣ E2 / ␻ 10R 8 ⬃⌬E a⫹b / ␻ 10RⰆ⌬E a⫹b and can be
neglected in the long distance limit.
As an aside we note that it is also straightforward to include the effects of magnetic polarizability; since
⑀ˆ 1 , ⑀ˆ 2
共44兲
The integration is performed in Appendix A, yielding
␻ 10RⰇ1
冕
2k̂ 1 "k̂ 2
28␲ 3
⫽⫺ 7 ,
k 1 ⫹k 2
R
as shown in Appendix A.
Here the fact that the large distance–retardation
corrected–van der Waals interaction varies as ␣ E2 /R 7 is easily understood from simple physical arguments, as cited by
Kaplan.14 In Figs. 2共a兲 and 2共b兲 the photon emission/
absorption involving, say, atom #1 occur entirely before 共or
after兲 those involving atom #2, so this interaction may effectively be shrunk to a local two-photon vertex characterized
by the electric/magnetic polarizability. Since polarizabilities
have units of volume and since two such interactions are
involved, the requirement that the potential energy have the
proper units gives V(R)⬃ ␣ E2 /R 7 , where we have used the
feature that the separation R provides the only scale in the
problem.
The above derivation breaks down, however, for short distances where the photon emission/absorption by the atoms is
correlated so that the photon emitted from atom #2, say, is
absorbed by atom #1 before any emission/absorption of the
second photon. Indeed, if we assume that ␻ 10RⰆ1, then it is
easy to see that the dominant diagrams become Figs. 2共c兲
and 2共d兲 with
⌬E c⫹d ⫽⌬E e⫹ f
⫻
共43兲
冕
d 3 k 1 d 3 k 2 e i 共 k1 ⫹k2 兲 "R
共 1⫹ 共 k̂ 1 "k̂ 2 兲 2 兲
.
k 1 ⫹k 2
共47兲
By simple scaling arguments we find that ⌬E a⫹b ⫽⌬E e⫹ f
2
⬃ ␣ E2 ␻ 10
/R 5 ⬃ ␻ 10R⌬E c⫹d Ⰶ⌬E c⫹d and can be neglected in
the short distance limit.
The general form of the van der Waals interaction at arbitrary distance can, of course, be found by adding together all
of the above diagrams,13 which yields the Casimir–Polder
integral
V 共 R 兲 ⫽⫺
4
␣ E2 ␻ 10
␲R3
冕
⬁
0
du exp共 ⫺2u 兲
2 2
R ⫹u 2 兲 2
共 ␻ 10
⫻ 共 u 4 ⫹2u 3 ⫹5u 2 ⫹6u⫹3 兲 .
Barry R. Holstein
共48兲
445
However, the physics is more clearly identified by working
in the separate long and short distance limits, as shown
above. If intermediate distance results are needed, a good
numerical match to the exact form is given by15
冉冊
3 ␻ 10␣ E2 2
23
V 共 R 兲 ⫽⫺
tan⫺1
.
4R 6
␲
6 ␻ 10R
共49兲
It is also interesting to see why the Kaplan argument leading to 1/R 7 behavior breaks down at short distance. The
point is that since emission/absorption by atoms #1, #2 are
correlated, it is not possible to shrink the interactions involving a single atom down to an effective point vertex involving
the polarizability. Indeed, in this case one must deal with
both the ground and excited states of both atoms, so that a
new scale ␻ 10 enters the problem and this is why simple
Kaplan scaling fails.
B. Two-photon exchange
An alternative, but completely equivalent, way to derive
the van der Waals interaction is to use the feature that the
potential is the Fourier transform of the quantum mechanical
scattering amplitude,15
V 共 R兲 ⫽
冕
d 3 q iq"R
e
amp共 q兲 ,
共 2␲ 兲3
共50兲
where q⫽pi ⫺p f is the momentum transfer. Thus, for example, for the Coulomb scattering amplitude ampCoul
⫽e 2 /q2 we find
V Coul共 R兲 ⫽e 2
冕
3
2
d q iq"R 1
e
,
3e
2⫽
q
4 ␲ 兩 R兩
共2␲兲
共51兲
as required. In our case, we wish to isolate the portion of the
atom–atom scattering which is due to two-photon exchange.
This is, in general, a nontrivial relativistic Feynman diagram
calculation, but may be simplified by realizing that the scattering amplitude is causal 共i.e., the scattered wave cannot be
emitted before the incoming plane wave strikes the scattering
center兲 and is an analytic function in momentum transfersquared, meaning that it satisfies a dispersion relation in the
variable t⫽⫺q2 ,
amp2 ␥ 共 t 兲 ⫽
1
␲
冕
⬁
0
dt Im amp2 ␥ 共 t ⬘ 兲
.
t ⬘ ⫺t⫺i ⑀
共52兲
The full discussion from this point onward is a bit technical
and is outlined in Appendix B. However, it is possible to
isolate the basic physics more simply and that is what we
shall present here. Readers wishing a more complete discussion can refer to Appendix B or to Ref. 15.
The desired van der Waals potential is found by taking the
Fourier transform of the two-photon-exchange amplitude. By
reversing the order of integrations we see that the result can
also be written in terms of a superposition of Yukawa potentials,
1
V共 R 兲⫽
␲
冕
⬁
0
1
⫽
4␲2
dt ⬘ Im amp2 ␥ 共 t ⬘ 兲
冕
⬁
0
冕
The necessary input to Eq. 共53兲 is provided by the unitarity
stricture, which in the center-of-mass frame takes the form16
446
Am. J. Phys., Vol. 69, No. 4, April 2001
兺 k兺, ⑀ˆ
2
兩 M 2 ␥ 共 t⫽ 共 k 1 ⫹k 2 兲 2 兲 兩 2
2
⫻ 共 2 ␲ 兲 ␦ 共 Q⫺k 1 ⫺k 2 兲 ,
共54兲
where the factor 2! is due to the identity of the two photons
in the intermediate state and Q ␮ represents the four-vector
( 冑t,0). Note here that the unitarity relation gives the imaginary component of the amplitude M 2 ␥ at positive values of t,
which can be related by crossing symmetry and analytic continuation to the Compton scattering amplitude at negative t.
Defining t⫽(2 ␻ ) 2 , we can write
V共 R 兲⫽
1
␲2
冕 ␻⬘
⬁
0
d␻⬘
exp共 ⫺2 ␻ ⬘ R 兲
Im amp2 ␥ 共 i ␻ ⬘ 兲 .
R
共55兲
The electric polarizability contribution to the Compton scattering amplitude from a neutral object can be written in the
general form
M Comp⯝4 ␲ ␣ E 共 ␻ 兲 E2 共 ␻ 兲 ⫽4 ␲ ␣ E 共 ␻ 兲 ␻ 2 ⑀ˆ 1 "⑀ˆ 2 ,
共56兲
where ␣ E ( ␻ ) is the dynamic polarizability. Thus the van der
Waals potential assumes the form
V 共 R 兲 ⯝⫺
冕 ␻⬘
⬁
4
3␲
5
0
d ␻ ⬘ ␣ E2 共 i ␻ ⬘ 兲
exp共 ⫺2 ␻ ⬘ R 兲
. 共57兲
R
Now suppose that the time taken by light to travel between
the two systems ( ␦ t 2 ⬃R) is large compared to a typical
excitation time ␦ t 1 ⬃1/␻ 10 . In this case ( ␻ 10RⰇ1) the integration in Eq. 共57兲 involves only values ␻ ⬘ ⭐1/RⰆ ␻ 10 .
Then ␣ E (i ␻ ) may be approximated by its value at ␻ ⫽0 and,
after five integrations by parts, the van der Waals potential
can be written in the form
␻ 10RⰇ1
V共 R 兲 → ⫺
⯝⫺
5! ␣ E2 共 0 兲
24␲
20␣ E2 共 0 兲
7
4␲R
冕
⬁
0
d␻⬘
exp共 ⫺2 ␻ ⬘ R 兲
R6
共58兲
.
Here the coefficient sitting in front of the polarizability is not
quite correct since we have not used the proper relativistic
forms 共cf. Appendix B兲, but the ‘‘physics’’ is properly included.
In the alternative limit where the light transition time is
small compared to a typical excitation time ( ␻ 10RⰆ1), the
convergence of the integration in Eq. 共57兲 is set now by ␻ ⬘
⭐ ␻ 10Ⰶ1/R and we must include the frequency dependence
of the dynamic polarizability. The correct way to do this can
be found in Appendix B, but one can understand the essential
physics of this result by inserting ␣ E2 (i ␻ ⬘ ) in Eq. 共58兲,
␻ 10RⰆ1
共53兲
1
2! k1 , ⑀ˆ 1
4 4
5!
V 共 R 兲 ⫽⫺
24␲
d 3q
e iq"R
共 2 ␲ 兲 3 t ⬘ ⫹q2 ⫺i ⑀
exp共 ⫺ 冑t ⬘ R 兲
dt Im amp2 ␥ 共 t ⬘ 兲
.
R
2 Im amp2 ␥ 共 t 兲 ⫽
→ ⫺
冕
⬁
0
␣ E2 共 i ␻ ⬘ 兲
d␻⬘
exp共 ⫺2 ␻ ⬘ R 兲
R6
5!
24␲ R 6
冕
⬁
0
d ␻ ⬘ ␣ E2 共 i ␻ ⬘ 兲 .
共59兲
Now the polarizability can itself be written in terms of a
dispersion relation17
Barry R. Holstein
446
␣ E共 ␻ 兲 ⫽
冕
d ␻ ⬘ ␻ ⬘␴共 ␻ ⬘ 兲
,
␻ ⬘2⫺ ␻ 2
␻0
1
␲
⬁
共60兲
where the spectral density ␴共␻⬘兲 is nonzero only in the vicinity of excitations of the system in question. As a simple
representation, we may write, then, for the case of a single
such excitation 关note that ␣ E (0) corresponds to ␣ E in the
notation of the previous section兴
␣ E共 ␻ 兲 ⯝ ␣ E
2
␻ 10
2
␻ 10
⫺␻2
共61兲
.
Substituting into Eq. 共59兲 we have
␻ 10RⰆ1
V共 R 兲 → ⫺
⫽⫺
冕
5! ␣ E2 共 0 兲
24␲ R 6
⬁
0
5! ␲ ␣ E2 共 0 兲 ␻ 10
96R 6
d␻⬘
tween the two forms arises because of the requirement to
utilize the static versus dynamic polarizability in the long
versus short distance limits. Either discussion involves interesting pieces of physics and offers an attractive means by
which to enhance discussion of this universal and familiar
phenomenon.
ACKNOWLEDGMENTS
It is pleasure to acknowledge useful conversations with
Dmitri Kharzeev, Lowell Brown, and David Kaplan as well
as the hospitality of the Institute for Nuclear Theory and the
Department of Physics at the University of Washington
where this paper was written. I also wish to thank Peter
Milonni for a helpful communication. This work is supported
in part by the National Science Foundation.
4
␻ 10
APPENDIX A
2 2
兲
共 ␻ ⬘ 2 ⫹ ␻ 10
In order to perform the integrals
共62兲
,
IV. CONCLUSIONS
In this paper we have examined the origin of the van der
Waals interaction, both in its traditional London—
1/R 6 —form, valid when the photon travel time between atoms is small compared to a typical quantum mechanical evolution time ⌬T⬃1/␻ 10 , and the retarded or Casimir–
Polder—1/R 7 —version, valid in the opposite limit when the
quantum mechanical evolution time is small compared to the
time required for interatomic photon travel so that finite
propagation times must be considered. The calculation was
performed in two very different but equivalent ways. The
first was a direct evaluation of the atom–atom interaction in
fourth-order perturbation theory, where the two limiting
forms are seen to arise from the dominance of differing
graphs depending on the interatomic separation. The second
was based on dispersion relations wherein the difference be-
J 关 1;2 兴 ⫽ 共 2 ␲ 兲 2
冕 ␴冕
⬁
⬁
d
0
0
dk 1 k 31
冕
⬁
0
dk 2 k 32
冕 冕
1
⫺1
dz
1
⫺1
共 4␲ 兲2
R7
⫽
共 4␲ 兲2
R7
⫹
⬁
⬁
d
0
0
⬁
⬁
d
0
0
ds 1 s 31
ds 1 s 31
⫻
关 1⫹ 共 k̂ 1 "k̂ 2 兲 2 ;2k̂ 1 "k̂ 2 兴
,
k 1 ⫹k 2
冕
冕
⬁
0
⬁
0
ds 2 s 32
ds 2 s 32
冊
冋
冋
共63兲
we select the z axis to lie along the direction R̂ and use the
identities
k̂ 1 "k̂ 2 ⫽cos ␪ 1 cos ␪ 2 ⫹sin ␪ 1 sin ␪ 2 cos共 ␾ 1 ⫺ ␾ 2 兲 ,
1
⫽
k 1 ⫹k 2
冕
共64兲
⬁
0
d ␴ exp⫺ ␴ 共 k 1 ⫹k 2 兲 ,
where k̂ 1 ,k̂ 2 are chosen to lie along the directions ␪, ␾ and
␪⬘, ␾⬘, respectively. Performing the ␾, ␾⬘ integrations in Eq.
共63兲 and defining s i ⫽k i R, ␬ ⫽ ␴ R we find
冉
冉
冊
册
冊
1 d2
d2
d d sin s 1 sin s 2
3 3 d2 d2
; ⫺2
exp⫺ ␬ 共 s 1 ⫹s 2 兲
⫹
2
2⫹
2⫹
2 2 ds 1 ds 2 2 ds 1 ds 22
ds 1 ds 2 s 1
s2
3 sin s 1 sin s 2 3 sin s 1
sin s 1
cos s 1
⫹
⫺2 3 ⫹2 2 共 same with s 1 →s 2 兲
2 s1
s2
2
s1
s1
s1
1 sin s 1
sin s 1
cos s 1
s2
⫺2 3 ⫹2 2 sin ⫹ 共 same with s 1 ↔s 2 兲
2
s1
s
s1
s1
2
⫺2
447
冉
冉
冕 ␬冕
冕 ␬冕
d 3 k 1 d 3 k 2 k 1 k 2 e i 共 k1 ⫹k2 兲 "R
dz ⬘ exp共 ik 1 Rz⫹ik 2 Rz ⬘ 兲
⫻exp⫺ ␴ 共 k 1 ⫹k 2 兲关 23 ⫹ 23 z 2 z ⬘ 2 ⫺ 21 共 z 2 ⫹z ⬘ 2 兲 ;2zz ⬘ 兴
⫽
冕
J 关 1;2 兴 共 R 兲 ⫽
which is the expected London form.
冊
册
cos s 1 sin s 1
⫺ 2 共 same with s 1 →s 2 兲 exp⫺ ␬ 共 s 1 ⫹s 2 兲 .
s1
s1
Am. J. Phys., Vol. 69, No. 4, April 2001
共65兲
Barry R. Holstein
447
Defining13
We begin by decomposing the Compton scattering amplitude for the reaction A(p)⫹ ␥ (k)→A(p ⬘ )⫹ ␥ (k ⬘ ) into electric and magnetic components. Defining
cos x sin x
⫺ 2 ,
x
x
M 共 x 兲 ⫽sin x⫹N 共 x 兲 ⫽sin x⫹
共66兲
sin x
,
K 共 x 兲 ⫽cos x⫺
x
we find
J 1⫽
2共 4␲ 兲2
R7
⫹
冉冕
⬁
0
⫺k ⬘ • Pk ␯ P ␮ 兲 /M A2 ,
冕 ␬冋 冉 冕
⬁
d
⬁
2
0
0
ds s 2 N 共 s 兲 e ⫺ ␬ s
ds s 2 M 共 s 兲 e ⫺ ␬ s
2共 4␲ 兲2
J 2⫽
R7
冕 ␬冉 冕
⬁
⬁
d
0
0
冊册
冊
2
where P⫽ p⫹ p ⬘ , it is easy to see that the corresponding
electric and magnetic tensors can be written as
2
T ␮E ␯ ⫽⫺ 21 T ␮1 ␯ ,
,
ds s K 共 s 兲 e
2
⫺␬s
冊
共67兲
2
.
冕
冕
0
⬁
0
n
共68兲
dn
␬
ds s cos s exp⫺ ␬ s⫽ 共 ⫺ 兲
,
d ␬ n 1⫹ ␬ 2
n
n
the remaining integrations over ␬ become trivial and we find
J 关 1,2兴 ⫽
关 23,⫺7 兴 4 ␲ 3
.
R7
冕
d 3 k 1 d 3 k 2 e i 共 k1 ⫹k2 兲 "R共 1⫹ 共 k̂ 1 "k̂ 2 兲 2 兲
共70兲
冕
⬁
0
d ␬ i exp⫺ ␬ i k i
冋 冉冕 冕
2共 4␲ 兲2
2
L 1⫽
R6
⫹
⬁
冉 冕 ␬冕
⬁
0
⬁
d
0
0
d␬
⬁
0
ds s N 共 s 兲 e
2
ds s 2 M 共 s 兲 e ⫺ ␬ s
冊册
⫺␬s
冊
2
⫻
24␲ 4
L 1⫽ 6 .
R
ds ⬘ ␳ E,M 共 s ⬘ ,t 兲
冊
1
1
⫹
.
s ⬘ ⫺s ⫹ s ⬘ ⫺u ⫹
共77兲
兺
X,Y ⫽E,M
M2
冕
ds ds ⬘ ␳ X 共 s,t 兲 ␳ Y 共 s ⬘ ,t 兲
⫻⌽ XY 共 s,s ⬘ ,t 兲 ,
共78兲
where
⌽ XY 共 s,s ⬘ ,t 兲 ⫽
共72兲
冕
共73兲
⫻
d 4k
d 4k ⬘
共 2 ␲ 兲 4 2k 0 共 2 ␲ 兲 4 2k 0⬘
The fully relativistic calculation of the van der Waals interaction via the Fourier transform method can be found in
the work of Feinberg and Sucher.15 We have tried in Sec.
III B to identify the basic physics behind this result. However, a number of the steps are less than rigorous. Here we
present a more detailed summary of how the calculation is
performed.
冉
1
1
⫹
s⫺s ⫹ s⫺u ⫹
冊冉
冊
1
1
⫹
.
s ⬘ ⫺s ⫺ s ⬘ ⫺u ⫺
共79兲
In order that t be positive and small it is necessary to do an
analytic continuation to consider the reactions A⫹Ā→Ā
⫹A at a total center-of-mass energy 冑t. In order that particles A, Ā stay on their mass shells we must use
冉冑 冊
冉冑 冊
APPENDIX B
Am. J. Phys., Vol. 69, No. 4, April 2001
M2
The amplitude Im amp2 ␥ (t) needed for the Fourier transform
in Eq. 共53兲 is then given by unitarity as
2
.
冉
⬁
⫻ 共 2 ␲ 兲 4 ␦ 4 共 t⫺k⫺k ⬘ 兲 T X :T Y
Using Eq. 共68兲 the integrations are again straightforward,
yielding
448
冕
1
␲
F E,M 共 s ⫹ ,t 兲 ⫽
共71兲
then by very similar steps to those used in deriving Eq. 共67兲
we find
共76兲
where we have defined s ⫾ ⫽(p⫾k) 2 , u ⫾ ⫽(p⫿k ⬘ ) 2 , and t
⫽(k⫺k ⬘ ) 2 . Using crossing symmetry, one now writes
F E ,F M in terms of a dispersion relation at fixed momentum
transfer,
Im amp2 ␥ ⫽⫺
may be performed. If we write
k 2i ⫽k 3i
共75兲
Then the general Compton scattering amplitude can be written as
共69兲
Likewise the corresponding short distance integration
L 1共 R 兲 ⫽
T ␮M␯ ⫽⫺ 21 T ␮1 ␯ ⫹2T ␮2 ␯ .
E
M
A ␮Comp
␯ ⫽T ␮ ␯ F E 共 s ⫹ ,t 兲 ⫹T ␮ ␯ F M 共 s ⫹ ,t 兲 ,
dn
1
ds s sin s exp⫺ ␬ s⫽ 共 ⫺ 兲
,
n
d ␬ 1⫹ ␬ 2
n
共74兲
T ␮2 ␯ ⫽k•k ⬘ g ␮ ␯ ⫺k ␯ k ␮ ,
Using
⬁
T ␮1 ␯ ⫽ 共 k•k ⬘ • Pg ␮ ␯ ⫹k•k ⬘ P ␮ P ␯ ⫺k• Pk ␮⬘ P ␯
P A,A¯ ⫽
t
,⫾iM A p̂ ,
2
⬘ ¯⫽
p A,A
冉冑
冊
t
,⫾iM A p̂ ⬘ ,
2
共80兲
t
k,k ⬘ ⫽
,⫾k̂ .
2
Then
Barry R. Holstein
448
t
s⫺s ⫺ ⫽s⫺M A2 ⫹ ⫺i 冑tM A z,
2
C XY 共 R 兲 ⫽
冕
⬁
0
d ␨ e ⫺2 ␨ R P XY 共 ␨ R 兲 ␣ E2 共 i ␨ 兲
共87兲
with
t
s⫺u ⫹ ⫽s⫺M A2 ⫹ ⫹i 冑tM A z,
2
s ⬘ ⫺s ⫹ ⫽s ⬘ ⫺M A2 ⫹
1
␲
P EE 共 x 兲 ⫽ P M M 共 x 兲 ⫽3⫹6x⫹5x 2 ⫹2x 3 ⫹x 4 ,
共81兲
t
⫹i 冑tM A z,
2
P EM 共 x 兲 ⫽ P M E 共 x 兲 ⫽⫺ 共 x 2 ⫹2x 3 ⫹x 4 兲 .
共88兲
The integration over ␨ may now be done, yielding results
identical to those found via perturbative methods.
t
s ⬘ ⫺u ⫺ ⫽s ⬘ ⫺M A2 ⫹ ⫺i 冑tM A z,
2
a兲
Electronic mail: holstein@physics.umass.edu
See, e.g., E. Merzbacher, Quantum Mechanics 共Wiley, New York, 1998兲.
2
H. Margenau and N. R. Kestner, Theory of Intermolecular Forces 共Pergamon, New York, 1969兲.
3
See, e.g., D. Park, Introduction to the Quantum Theory 共McGraw–Hill,
New York, 1974兲, Chap. 16.4. See, especially, P. W. Milonni, The Quantum Vacuum: An Introduction to Quantum Electrodynamics 共Academic,
New York, 1994兲; E. A. Power, Introductory Quantum Electrodynamics
共Longmans, London, 1964兲.
4
This interaction between two polarizable systems is sometimes called the
‘‘dispersive’’ van der Waals force, since it is associated with the atomic
polarizabilities which determine the index of refraction. This is related to,
but distinct from, the ‘‘orientation’’ van der Waals interaction between
two systems both of which have a permanent dipole moment and the
‘‘inductive’’ interaction between a polarizable and a polarized system.
5
C. Itzykson and J.-B. Zuber, Quantum Field Theory 共McGraw–Hill, New
York, 1980兲.
6
H. Fujii and D. Kharzeev, ‘‘Long-range Forces of QCD,’’ Phys. Rev. D
60, 114039, 1–12 共1999兲. In this case the authors show that the long
distance component of the interaction 共which is the QCD analog of the van
der Waals interaction in atomic physics兲 is due to the exchange of a pair of
pions and leads to an interaction of the form V(R)⬃ 关 exp(⫺2m␲R)兴/R.
7
F. London, ‘‘Zur Theorie und Systematik der Molekularkräfte,’’ Z. Phys.
63, 245–279 共1930兲.
8
H. B. G. Casimir and D. Polder, ‘‘The Influence of Retardation on the
London–van der Waals Force,’’ Phys. Rev. 73, 366–372 共1948兲.
9
C. Kittel, Introduction to Solid State Physics 共Wiley, New York, 1996兲,
Chap. 3.
10
See, e.g., Ref. 1, Chap. 18.
11
C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics 共Wiley,
New York, 1977兲, Chap. XI.
12
See, e.g., B. R. Holstein, Topics in Advanced Quantum Mechanics
共Addison–Wesley, Reading, MA, 1992兲, Chap. 3.
13
E. A. Power and S. Zienau, ‘‘On the Radiation Contribution to the van der
Waals Force,’’ Nuovo Cimento 6, 7–17 共1957兲.
14
D. Kaplan 共private communication兲.
15
G. Feinberg and J. Sucher, ‘‘General Theory of the van der Waals Interaction: A Model-independent Approach,’’ Phys. Rev. A 2, 2395–2415
共1970兲.
16
In the forward direction this is simply the familiar optical theorem.
17
See, e.g., B. R. Holstein, ‘‘Electromagnetic Polarizability of the
Nucleon,’’ Comments Nucl. Part. Phys. 20, 301–324 共1992兲.
1
where z⫽ p̂"k̂⫽⫺p̂ ⬘ "k̂. We find then
1
⌽ XY ⯝
2
⫻
冕
冉
d⍀T X :T Y
冉
s⫺M A2
共 s⫺M A2 兲 2 ⫹tM A2 z 2
s ⬘ ⫺M A2
共 s ⬘ ⫺M A2 兲 2 ⫹tM A2 z ⬘ 2
冊
冊
共82兲
.
Defining ␻ ⫽(s⫺M A2 )/M A , ␻ ⬘ ⫽(s ⬘ ⫺M A2 )/M A , and ␶
⫽ ␻ / 冑t, ␶ ⬘ ⫽ ␻ ⬘ / 冑t the solid angle integration becomes
⌽⯝
␲t
2M A2
冕
1
⫺1
dz
␶
␶⬘
⌰ ,
2
2
␶ ⫹z ␶ ⬘ ⫹z 2 XY
2
共83兲
where ⌰ EE ⫽⌰ M M ⫽2⫺2z 2 ⫹z 4 , ⌰ EM ⫽⌰ M E ⫽z 4 ⫺2z 2 ,
and may be performed directly, yielding
⌽ XY ⫽
␲ t ␶␶ ⬘ g XY 共 ␶ ⬘ 兲 ⫺g XY 共 ␶ 兲
␶ 2⫺ ␶ ⬘2
M A2
with
冉
冉
共84兲
冊
2⫹2 ␶ 2 ⫹ ␶ 4
tan⫺1 ␶ ,
g EE 共 ␶ 兲 ⫽g M M 共 ␶ 兲 ⫽ ␶ ⫺
␶
2
冊
2 ␶ 2⫹ ␶ 4
g EM 共 ␶ 兲 ⫽g M E 共 ␶ 兲 ⫽ ␶ ⫺
tan⫺1 ␶ .
␶
2
共85兲
Using these results the integration in Eq. 共53兲 can be performed by making the substitution t⫽4 ␨ 2 followed by a
fivefold integration by parts in order to remove the tan⫺1 ␶
dependence. The result is
V 共 R 兲 ⫽⫺
C XY 共 R 兲
,
R6
where
449
Am. J. Phys., Vol. 69, No. 4, April 2001
共86兲
Barry R. Holstein
449