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`Bound' states of an electron in the far-field of a polar molecule
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1996 Eur. J. Phys. 17 275
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275
Eur. J. Phys. 17 (1996) 275–278. Printed in the UK
‘Bound’ states of an electron in the
far-field of a polar molecule
G P Sastry, V Srinivas and A V Madhav†
Department of Physics and Meteorology, Indian Institute of Technology, Kharagpur 721 302, India
Received 29 March 1996
Abstract. The classical and quantum mechanics of an
electron moving in the field of an electric dipole is studied in
some detail. It is shown that there exists a family of circular
orbits lying on the surface of a half-cone stretching from the
site of the dipole to infinity. All these orbits have the same
energy and angular momentum. Although the dipole potential
is highly non-central, the Schrödinger equation turns out to
be separable in spherical polar coordinates. It is shown that
the cone of the classical ‘bound’ states also develops from
the angular part of the Schrödinger equation on imposing the
Wilson–Sommerfeld quantization rule. Instead of a discrete
spectrum of energy eigenvalues, we now have a discrete
spectrum of dipole moments for which such ‘bound’ states
are at all possible. All these conclusions, which are in accord
with the virial theorem, reduce to the results of Lévy-Leblond
and Balibar obtained from the Heisenberg inequalities
ignoring the angular dependence of the dipole potential.
Zusammenfassung. Die Klassische und Quantenmechanik
von einem Elektron, das im Feld Von einem elektrischen
Dipol bewegt wird hier ansfürlich studiert. Es wird gezeight:
es gibt eine Familie der Kriesförmigen Bahnen, die auf der
Oberfläche von einem Halb-Kegel liegt und streckt von der
Seite des Dipols bis Unendlichkeit. Alle diese Bahnen haben
die gleichen Energie und das Drehmoment. Obwohl das
Dipolpotential ganz nicht-zentral ist, wird die Schrödinger
Gleichung in Kugelpolar Koordinaten trennbar. Es wird
gezegeit, dass der Kugel von dem klassichen ‘Aufprall’
Zustände auch von dem winkelförmigen Teil der Schrödinger
Gleichung entwickelt und drückt Wilson–Sommerfeld
Quantelung Regel ein. Statt des diskreten Spektrums von
Energie Eigenwert, haben wir jetzt diskretes Spektrum von
Dipol Momenten, wofür solche ‘Aufprall’ Zustände immer
möglich sind. Alle diese Abschlüsse nach virial Theorem hat
dasselbe Ergebnis wie Lévy-Leblond und Balibar, das ohne
winkelförmige Abhängigkeit von dem Dipolpotential, von
den Heisenberg Ungleicheiten erhalten wurde.
1. Introduction
is an involved but fairly standard type of problem in
quantum chemistry. However, an electron in the farfield of such a molecule sees it as a point dipole (LévyLeblond and Balibar 1990). Similar situations may also
arise in defect-state ferroelectrics.
However, our interest in this problem arose out of a
question on the separability of the Schrödinger equation
in spherical polar coordinates. In contrast to popular
notions, the Schrödinger equation in the non-central
dipole potential separates in spherical polar coordinates.
Further, the separated equations look so good that they
admit interesting conclusions.
Bertrand’s theorem in classical mechanics (Goldstein
1980) states that the only central potentials which admit
stable closed orbits are those belonging to the inverse
square force (the Kepler problem) and the parabolic
potential (the oscillator problem). In the non-central
field of a dipole, we show that there exists a family
of degenerate circular ‘knife-edge’ orbits lying on the
surface of a half-cone. These have interesting energy
and angular momentum properties which are preserved
in the quantum mechanical equations. We present
them here in the hope that they may be of some
The motion of an electron in the field of a point charge
is undoubtedly the most celebrated problem in atomic
physics (the hydrogen atom), but the motion of an
electron in the field of an electric dipole does not seem
to have been studied at all. There is indeed good reason
for this: no nucleus has a measurable electric dipole
moment. Nuclei do have magnetic dipole moments, but
there are no magnetic monopoles to go round them. This
is why the next important problem in atomic physics
turns out to be the dipole–dipole interaction: in its
electrical version it gives rise to the van der Waals forces
and in its magnetic version to hyperfine splitting.
Polar molecules, such as the water molecule, do have
large electric dipole moments due to their asymmetrical
electron clouds. An extra electron moving near such
a polar molecule sees all the nuclei and their bound
electrons. The resulting Hamiltonian is simply that for
the negative ion of the molecule, e.g. (H2 O)− , which
† Present address:
Department of Physics, Jadwin Hall,
Princeton University, Princeton, NJ 08544, USA
c 1996 IOP Publishing Ltd & The European Physical Society
0143-0807/96/050275+04$19.50 G P Sastry et al
276
interest to students and teachers of classical and quantum
mechanics.
The energy of the electron in its circular orbit is given
by
E = mv 2 /2 − (ed cos θc )/r 2 .
2. The classical orbits
Posting the values of v and θc from equations (8) and
(3) in equation (9), we get
Let us fix the electric dipole at the origin of the spherical
polar coordinates and align the z-axis along the dipole
moment vector p (figure 1). The electrostatic scalar
potential U (r, θ ) of the dipole field is given by
U = (d cos θ)/r 2 ,
(1)
d = p/(4π 0 ).
(2)
where
It can be checked that there exists a cone about the
dipole axis with the semi-vertical angle
√
θc = tan−1 ( 2),
(3)
at every point on which the z-component of the electric
field Ez vanishes identically (figure 1). All the electric
field lines of the dipole take U-turns when they arrive at
the surface of this cone. The electric field at all points
on this cone is perpendicular to the dipole axis, and has
the value
Eρ = (3d sin θc cos θc )/r 3 .
Inserting equation (3) in equation (4), we get
√
Eρ = ( 2d)/r 3 .
(4)
(5)
Since the azimuthal angle φ is cyclic in the dipole
potential, the z-component of the angular momentum,
Lz , of an electron moving in the dipole field is a constant
of the motion. An electron released from rest at any
point on the upper half of the z-axis falls into the dipole,
while an electron released from rest at any point on the
lower half of the z-axis is repelled to infinity.
However, consider an electron at a point (r, θc ) on the
surface of the upper half-cone (figure 1). The electrical
force acting on the electron is entirely centripetal, i.e.
towards the z-axis. If we project this electron with an
appropriate speed v tangential to the cone perpendicular
to the dipole axis (i.e. in the eφ direction), we can make
the electron go round in a circular orbit of radius
ρ = r sin θc
(6)
lying on the surface of the cone parallel to its base. The
required speed v is given by the condition
mv /ρ = eEρ ,
2
(9)
(7)
where e and m are the magnitudes of the charge and the
mass of the electron. From equations (5), (6) and (7),
we get
√
v = (2ed/ 3m)1/2 /r.
(8)
The higher up the cone the electron goes, the smaller
is the required speed v. The rotation can be clockwise
or counterclockwise. No such orbits are possible on
the lower half-cone where the force on the electron is
centrifugal.
E = 0.
(10)
This is in accordance with the virial theorem for ‘bound’
states of finite motion (Goldstein 1980) which states that
2hT i = hr(∂V /∂r)i,
(11)
where T and V are the kinetic and the potential energies
of the electron. For the dipole potential given by
equation (1), equation (11) gives
hT i = −hV i
(12)
whence
hEi = hT i + hV i = 0.
(13)
If the speed of the electron v falls below the critical
value required by equation (8), the energy E given by
equation (9) becomes negative and the electron falls
into the dipole. If v exceeds the critical value, E turns
positive and the electron runs away to infinity. Thus,
there exists an infinity of ‘knife-edge’ circular orbits of
all possible radii lying on the upper half-cone, all of
which have exactly zero energy.
More interesting is the angular momentum Lz of the
electron in a circular orbit of radius ρ:
Lz = mvρ.
From equations (6) and (8) we get
√
Lz = (4edm/ 27)1/2 .
(14)
(15)
Note that r has disappeared from Lz of equation (15).
Hence all the circular orbits have the same angular
momentum Lz . The inverse square potential is special
in this property. Other central potentials also admit
circular orbits but they have different energies and
angular momenta depending on their radii. Classically,
there is no restriction on Lz of equation (15) and the
electron can move in any one of these circular orbits for
arbitrary values of the dipole moment p.
3. Quantum considerations
The Hamiltonian operator for the electron in the field of
the electric dipole is given by
H = −(h̄2 /2m)∇ 2 − (ed cos θ)/r 2 .
(16)
Although the Hamiltonian in equation (16) contains a
non-central potential, the time-independent Schrödinger
equation
H ψ(r, θ, φ) = E ψ(r, θ, φ) ≡ ER(r)2(θ)8(φ)
(17)
separates in spherical polar coordinates by virtue of
the r 2 term in the denominator of the dipole potential.
An electron in the far-field of a polar molecule
277
Figure 1. The electron orbit in the far-field of a polar molecule: (1) electric dipole, (2) cone of circular orbits,
(3) electric field line and (4) electron orbit.
Any other power of r going with cos θ would spoil the
separability.
Since φ is cyclic, Lz commutes with the dipole
Hamiltonian (16) and the azimuthal equation has the
usual form
d2 8(φ)
+ m2l 8(φ) = 0,
(18)
dφ 2
with the solution
8(φ) = exp(iml φ).
(19)
The single-valuedness of 8(φ) restricts the quantum
number ml to the spectrum
(20)
ml = 0, ±1, ±2, . . . .
Since 8(φ) is an eigenfunction of the Lz operator,
(21)
(Lz )op 8(φ) = h̄ml 8(φ),
the eigenvalues of Lz are given by
(22)
Lz = ml h̄.
The angular equation for 2(θ ) is given by
d2(θ)
1 d
sin θ
sin θ dθ
dθ
2med cos θ
m2l
+ C+
2(θ ) = 0, (23)
−
h̄2
sin2 θ
where C is the second separation constant. If the dipole
moment d were equal to zero, equation (23) would be
identical to the associated Legendre equation, with C
restricted to l(l + 1), l = 0, 1, 2, . . . , −l ≤ ml ≤ l, and l
having the significance of the total angular momentum
quantum number. For the dipole potential d 6= 0, L2
does not commute with the Hamiltonian and the total
angular momentum is not a constant of the motion. With
d 6= 0, the presence of the cos θ term in equation (23)
makes it difficult to solve. It could not be reduced easily
to the differential equation of any well known special
function.
Nevertheless, we can bring out the cone of classical
orbits hidden in the angular equation (23). Using
equations (3), (15) and (22) and invoking the Wilson–
Sommerfeld quantization rule, we see that, on the cone
θ = θc , the two θ-dependent terms in the second
brackets of equation (23) cancel exactly. Therefore,
right on the cone where the probability density is
expected to be maximum, equation (23) reduces to the
Legendre equation with the solutions Pl (cos θc ) and
C = l(l + 1), l = 0, 1, 2 . . .. However, l here is
no longer the angular quantum number since L2 is no
longer a constant of the motion. On the other hand, ml
is the good quantum number that signifies the conserved
278
G P Sastry et al
angular momentum Lz . There is no longer the (2l + 1)fold degeneracy since there is no spherical symmetry in
the problem.
We thus have the result that, in the quantum domain,
the ‘bound’ states should satisfy the condition
√
(4edm)/ 27 = m2l h̄2
ml = 0, 1, 2, . . . .
(24)
wavefunctions would then be the spherical harmonics
with a (2l + 1)-fold degeneracy resulting from the
spherical symmetry. A quantum condition similar to
equation (25) would then arise from the radial equation
with E = 0.
The value ml = 0 corresponds to the uninteresting free
particle solution. On the other hand, the other values
of ml place severe restrictions on the magnitude of
the dipole moment for which these ‘bound’ states are
permitted. The smallest allowed value of p is therefore
given by
√
p = ( 27π 0h̄2 )/em.
(25)
4. Conclusions
Evaluation of p from equation (25) shows that it has
the same order of magnitude as the molecular dipole
moments occurring in nature. Similar estimates also
follow from the use of the Heisenberg inequalities
(Lévy-Leblond and Balibar 1990).
The radial equation for R(r) has the form
1 d
C
2mE
2 dR(r)
R(r) = 0.
(26)
−
r
+
r 2 dr
dr
r2
h̄2
The dipole potential has dropped away from the radial
equation (26). For ‘bound’ states, the quantum version
of the virial theorem (Merzbacher 1970), viz,
2hT i = hr · ∇V i
(27)
E = hEi = hT i + hV i = 0,
(28)
The quantum and classical mechanics of an electron
moving in the non-central field of a point electric dipole
is shown to exhibit some interesting features. It is shown
that there exists a family of circular orbits lying on the
surface of a half-cone with vertex at the dipole. Unlike
those for other central forces, all these orbits have
the same energy and angular momentum. Although it
involves non-central forces, the Schrödinger equation is
separable in spherical polar coordinates. The azimuthal
equation leads to quantization of Lz . The cone of
classical orbits makes its appearance in the angular
equation. The quantum condition restricts the permitted
values of the dipole moment for which such ‘bound’
states can exist. These permitted values agree with the
order-of-magnitude estimates obtained by Lévy-Leblond
and Balibar (1990) from the Heisenberg inequalities
ignoring the angular dependence of the dipole potential.
Acknowledgments
gives
which agrees with the classical result. With E = 0,
equation (26) shows that a ‘barely’ normalizable radial
wavefunction R(r) occurs for C = 0, i.e. l = 0.
These quantum considerations reveal the classical
orbits lying on the cone θ = θc , with the additional constraint that for molecular dipoles the dipole
moment p which permits these ‘bound’ states is restricted by the quantum condition equation (25). It
is instructive to compare the classical and quantum
features of this problem with those for an electron
moving in a central inverse square potential U ∝
r −2 , without the angular cos θ factor. The angular
We are deeply indebted to Professor K L Chopra for his
kind encouragement and to Professor Debabrata Basu
for helpful discussions. We thank Professor V N Giri
for the abstract in German.
References
Goldstein H 1980 Classical Mechanics 2nd edn (Reading,
MA: Addison-Wesley)
Lévy-Leblond J-M and Balibar F 1990 Quantics (Amsterdam:
North-Holland) pp 140–1
Merzbacher E 1970 Quantum Mechanics 2nd edn (New York:
Wiley)