Quantum mechanics of the 1∕x2 potential

Andrew M. Essin and David J. Griffiths

Citation: Am. J. Phys. 74, 109 (2006); doi: 10.1119/1.2165248

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Quantum mechanics of the 1 / x2 potential
Andrew M. Essina兲
Department of Physics, University of California, Berkeley, California 94720
David J. Griffithsb兲
Department of Physics, Reed College, Portland, Oregon 97202
共Received 19 September 2005; accepted 12 December 2005兲
In quantum mechanics a localized attractive potential typically supports a 共possibly infinite兲 set of
bound states, characterized by a discrete spectrum of allowed energies, together with a continuum
of scattering states, characterized 共in one dimension兲 by an energy-dependent phase shift. The 1 / x2
potential on 0 ⬍ x ⬍ ⬁ confounds all of our intuitions and expectations. Resolving its paradoxes
requires sophisticated theoretical machinery: regularization, renormalization, anomalous
symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a
level accessible to advanced undergraduates. © 2006 American Association of Physics Teachers.
关DOI: 10.1119/1.2165248兴

I. INTRODUCTION Hamiltonian.2 But let’s pretend for a moment that we did not
notice this problem. We look for bound states—normalizable
Ordinarily, an attractive potential admits discrete bound negative-energy solutions of the Schrödinger equation:
states, together 共perhaps兲 with a continuum of scattering
states. In a first course on quantum mechanics, students en- ប 2 d 2␺ a
counter the infinite square well, the harmonic oscillator, the − − ␺ = E␺ 共x ⬎ 0兲, 共2兲
2m dx2 x2
Dirac delta function, the finite square well, and 共in three
dimensions兲 the spherical well and the Coulomb potential, all or, multiplying through by −2m / ប2,
of which fit this paradigm 共though the first two lack scatter-
ing states兲. We do not study the 1 / x2 potential, and for good d 2␺ ␣
reason: It violates every rule in the book, and discredits all + ␺ = ␬ 2␺ , 共3兲
dx2 x2
the intuition we are trying to instill in our students. In spite
of this 共or rather, precisely because of it兲 the 1 / x2 potential is where ␣ ⬅ 2ma / ប2 and ␬ ⬅ 冑−2mE / ប, subject to the bound-
a fascinating system, and analyzing its paradoxes provides an ary conditions
illuminating introduction to some of the more subtle tech-
niques in contemporary theoretical physics: regularization ␺ → 0 as x → 0 and x → ⬁. 共4兲
and renormalization, anomalous symmetry-breaking, and
self-adjoint extensions. In this paper we tell the story from a The first boundary condition is necessary to make ␺ continu-
pedagogical perspective, starting out innocent and naive, and ous at the origin 共␺ = 0, of course, for x ⬍ 0兲; the second is
letting the unfolding saga force us to become wiser and more required for normalization:

冕
sophisticated. ⬁
In Sec. II we introduce the problem in its simplest 共one-
兩␺共x兲兩2dx = 1. 共5兲
dimensional兲 form, and approach it as we would any other 0
quantum system. We quickly encounter a series of puzzles
and surprises. In Sec. III we identify the source of the diffi- Suppose we could find just one bound state ␺共x兲, with
culties and modify the potential so as to avoid the trouble. energy E. Scaling x by a factor ␤, we can immediately con-
This leads naturally to renormalization and anomalies. In struct a new solution, ␺␤共x兲 ⬅ ␺共␤x兲, with energy ␤2E:
Sec. IV we notice that the Hamiltonian is not Hermitian, and
modify the space of permissible functions to make it so; this d 2␺ ␤ ␣ d2 ␣ 2 d
2

introduces the method of self-adjoint extensions. In Sec. V + ␺ ␤ = ␺ 共 ␤ x兲 + ␺ 共 ␤ x兲 = ␤ ␺共u兲

dx2 x2 dx2 x2 du2
we draw lessons from our experience and point to some real-
world applications. ␣
+ ␤2 ␺共u兲 = ␤2␬2␺共u兲 = 共␤␬兲2␺␤共x兲,
u2
II. PECULIARITIES OF THE 1 / x2 POTENTIAL 共6兲
Consider a particle of mass m in the one-dimensional where u ⬅ ␤x. But ␤ can be any real number.3 So if there
potential1

再 冎
exist any bound states, then there is a bound state for every
⬁ 共x 艋 0兲 negative energy! In particular, the 1 / x2 potential has no
V共x兲 = 共1兲 ground state. This is disturbing, for a system without a lower
− a/x 2
共x ⬎ 0兲
limit on its allowed energies would be wildly unstable, cas-
shown in Fig. 1. Here a is a constant with the 共MKS兲 units cading down with the release of an unlimited amount of
J m2. In truth, the alarm should already be sounding, for energy. A reasonable inference would be that there are no
there is no way to construct a quantity with the dimensions negative energy states, which is indeed the case for ␣
of energy from the parameters at hand 共ប, m, and a兲, and ⬍ 1 / 4 共trivially so, for ␣ ⬍ 0, because the potential is then
hence no possible formula for the eigenvalues of the repulsive兲.

109 Am. J. Phys. 74 共2兲, February 2006 http://aapt.org/ajp © 2006 American Association of Physics Teachers 109

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 Fig. 1. The 1 / x2 potential. Fig. 2. The bound state wave function, ␺␬共x兲.

But when ␣ ⬎ 1 / 4 there certainly do exist negative energy

One particularly nice way to show that there are no bound solutions. The Schrödinger equation can be solved by the
states for 0 ⬍ ␣ ⬍ 1 / 4 is to factor the Hamiltonian.4 Let ␣ method of Frobenius: We write ␺共x兲 as a power series,
= ␯共1 − ␯兲 define the new constant ␯, ⬁

H=−
ប2 d2
−
a
2m dx2 x2
= −
ប2 d ␯
+
2m dx x
冉 冊冉 d ␯
−
dx x
冊. 共7兲
␺共x兲 = xs 兺 a jx j
j=0
共a0 ⫽ 0兲, 共13兲

and substitute this into Eq. 共3兲:

For an arbitrary test function f共x兲 we have

冉 冊冉 冊 冉 冊冉 冊
⬁ ⬁
d ␯
+
d ␯
− f共x兲 =
d ␯
+
df ␯
− f 共8a兲 兺 a j关共j + s兲共j + s − 1兲 + ␣兴x j−2 = ␬2兺 a jx j . 共14兲
dx x dx x dx x dx x j=0 j=0

We equate like powers and find that the x−2 term yields s共s
d2 f ␯ ␯ df ␯ df ␯2 − 1兲 + ␣ = 0, so
= 2 + 2f − + − f 共8b兲
dx x x dx x dx x2
s = ␯ = 共1/2兲 ± 冑共1/4兲 − ␣; 共15兲
d f ␯共1 − ␯兲
= 2+
dx
2

x 2 f=
d
dx
␣
2 + 2 f.
x
冉 2
冊 共8c兲
−1
the x term forces a1 = 0, and the remaining coefficients are
determined by the recursion relation
Now, the Hermitian conjugate of 共d / dx + ␯ / x兲 is 共−d / dx ␬2
aj = a j−2 共j = 2,3,4, . . . 兲. 共16兲
+ ␯* / x兲:

冓 冏冉 冊冔 冕 冉 冊
j共j + 2s − 1兲
⬁
d ␯ dg ␯ There are two solutions—one for each sign in Eq. 共15兲. Near
f + g = f* + g dx 共9a兲 the origin they go like a0xs; for ␣ ⬎ 1 / 4 this means
dx x 0 dx x
冑xe±ig ln x ,
冏 冕冉 冊 冕冉 冊
共17兲
⬁ ⬁ ⬁
df *
␯* *
=f g*
− g dx + f g dx 共9b兲 where g ⬅ 冑␣ − 1 / 4.
0 0 dx 0 x The general solution is a linear combination, but it turns

冓冏冉 冊 冏 冔
out that only one combination is normalizable:
d ␯*
= − + f g , 共9c兲 ␺␬共x兲 = A冑xKig共␬x兲, 共18兲
dx x
where Kig is the modified Bessel function of order ig; ␺␬ is 5
provided that f共x兲 and g共x兲 go to zero at 0 and ⬁. Thus

冓 冏冉 冊冉 冊 冔
real, as long as g is real 共which is to say, for ␣ ⬎ 1 / 4兲, and
ប2 d ␯ d ␯ finite at the origin 共so ␺␬ → 0 as x → 0兲; ␺␬ itself is
E = 具H典 = 具␺兩H兩␺典 = − ␺ + − ␺ square-integrable:6
2m dx x dx x
共10a兲
冕 ⬁
兩冑xKig共␬x兲兩2dx =
␲g
, 共19兲

冓冉 冊 冏冉 冊 冔
0 2␬ sinh共␲g兲
2

ប2 d ␯* d ␯
= − ␺ − ␺ . 共10b兲 so the normalization constant is

冑
2m dx x dx x
2 sinh共␲g兲
If ␯ is real, then, A=␬ . 共20兲

冕 冏冉 冊冏
⬁
␲g
ប2
d␺ ␯ 2
E= − ␺ dx ⬎ 0, 共11兲 A plot of the wave function is shown in Fig. 2. Notice the
2m dx x
0 oscillations as x → 0, which result from the sinusoidal depen-
so negative energy states cannot occur if ␯ is real. But ␯共1 dence on g ln x in Eq. 共17兲. We are accustomed to the idea
− ␯兲 = ␣, so that the ground state has no zero crossings, the first excited
state has one, the second two, and so on. But the 1 / x2 po-
␯=
1
2
± 冑 1
4
− ␣, 共12兲
tential has no ground state, and every 共negative energy兲 so-
lution has an infinite number of zero crossings. If the number
of nodes counts the number of lower energy states, then no
and hence if ␣ ⬍ 1 / 4, then ␯ is real, and there can be no matter what the energy, there are always infinitely many
bound states. states even lower.7

110 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 110

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 What about positive energy 共scattering兲 states? For E ⬎ 0
the general solution to Schrödinger’s equation 关Eq. 共2兲兴 is
␺k共x兲 = 冑x关AH共2兲 共1兲
ig 共kx兲 + BHig 共kx兲兴, 共21兲
where k ⬅ 冑2mE / ប, and H共1兲 and H共2兲 are Hankel functions.
For large x,8

H共1兲
ig 共kx兲 ⬃ 冑 2 i共kx−␲/4兲 ␲g/2
␲kx
e e , 共22a兲

H共2兲
ig 共kx兲 ⬃ 冑 2 −i共kx−␲/4兲 −␲g/2
␲kx
e e , 共22b兲

so

冑 冋 册
Fig. 4. Ground state and first three excited states, as functions of ␬x, for V⑀,
2 −␲g/2 i␲/4 −ikx B ␲g ikx with ⑀ = 1 and g = 3 共not normalized兲.
␺k共x兲 ⬃ A e e e −i e e . 共23兲
␲k A

再 冎
The first term 共e−ikx兲 represents a wave incident from the
right; the second 共eikx兲 is the reflected wave. Ordinarily, set- ⬁ 共x 艋 ⑀兲
ting ␺k共0兲 = 0 in Eq. 共21兲 would determine B / A, and the V⑀共x兲 = 共25兲
− a/x 2
共x ⬎ ⑀兲,
asymptotic expression Eq. 共23兲 would reduce to9
as shown in Fig. 3. This “regularized” potential suffers none
␺k共x兲 ⬃ 关e−ikx − ei共2␦+kx兲兴, 共24兲 of the ills that afflict V共x兲; we propose to work with V⑀共x兲,
indicating that the reflected wave is equal in amplitude to the and take the limit ⑀ → 0 only at the very end. Of course, the
incident wave 共as required by conservation of probability兲 pathologies can be expected to reappear in this limit, but as
and shifted in phase by an amount ␦共k兲. But in this case we shall see, some predictions survive, and these we take to
be the “true” physical content of the 1 / x2 system. This is the
␺k共0兲 is automatically zero 共according to Eq. 共17兲, 兩␺兩 ⬃ 冑x兲. strategy of “renormalization.”
There is no constraint on B, no formula for ␦共k兲, and 共most Having introduced a parameter 共⑀兲 with the units of
alarming兲 no enforcement of conservation of probability; the length, we are now able to construct an expression with the
outgoing wave can have any amplitude! dimensions of energy:
Conclusion: The 1 / x2 potential has no ground state, and
the allowed energies are not quantized. As long as ␣ ⬎ 1 / 4, a
the Schrödinger equation can be solved 共and the boundary f共␣兲, 共26兲
⑀2
conditions satisfied兲 for every negative energy; the solutions
are real and normalizable, and each of them has an infinite where f is a function of the dimensionless quantity ␣
number of zero crossings. Scattering states occur for every = 2ma / ប2. The system is no longer scale invariant; it pos-
positive energy, but the boundary condition at x = 0 imposes sesses both a ground state11 and a discrete spectrum of bound
no constraint on the reflection coefficient, and does not de- states. These have the same functional form as in Eq. 共18兲,
termine the phase shift. It would be difficult to imagine a but the boundary condition is now ␺␬共⑀兲 = 0, which implies
situation more at odds with our expectations. that
Kig共␬⑀兲 = 0, 共27兲
which is not automatically satisfied, and serves to quantize
III. REGULARIZATION, RENORMALIZATION, AND
the energy. Figure 4 shows the ground state and the first three
ANOMALIES excited states for the case ␣ = 9.25 共g = 3兲.12 Figure 5 shows
Physically, the 1 / x2 potential is just too strong at the the ground state energy 共or rather, ␬1⑀兲 for g ranging up to 7.
origin—1 / x is acceptable,10 but 1 / x2 is not. One way to If ␬⑀ Ⰶ 1,13 we can provide a relatively simple formula for
avoid the problem is to move the “wall” over to the right a the allowed energies:14
distance ⑀:
Kig共z兲 ⬇ − 冑 ␲
g sinh共␲g兲

冋 冉冊
⫻sin g ln
z
2
− arg ⌫共1 + ig兲 册 共z Ⰶ 1兲, 共28兲

so Kig共␬⑀兲 = 0 implies

g ln 冉 冊␬⑀
2
− arg ⌫共1 + ig兲 + n␲ = 0, 共29兲

where n is an integer. The ground state is n = 1,15 and the

Fig. 3. The regularized potential. excited states are n = 2 , 3 , . . ., with

111 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 111

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 Fig. 6. Graphs of ␦ as a function of k⑀, for g = 0.5 共top兲 and g = 0 共bottom兲.
Fig. 5. The ground state ␬1⑀ as a function of g. The graphs on the right show the behavior near the origin 共the horizontal
scale is in powers of 10兲; they suggest that there is no limit when g = 0.5,
whereas ␦ → ␲ / 4 when g = 0 共though that limit is only approached when
k⑀ Ⰶ 10−10兲.

2
␬n = e关arg ⌫共1+ig兲−n␲兴/g , 共30兲
⑀
tan ␰ + tanh共␲g/2兲
or tan ␦ ⬇ , 共38兲
tan ␰ − tanh共␲g/2兲
2
2ប 2关arg ⌫共1+ig兲−n␲兴/g
En = − e . 共31兲 where
m⑀2
There are infinitely many discrete bound states, as one might
␰ ⬅ g ln共k⑀/2兲 − arg ⌫共1 + ig兲. 共39兲
expect.16 As k⑀ → 0, ␰ → −⬁, and tan ␰ fluctuates wildly: Whenever ␰
For positive energies 共scattering兲 the wave function is still hits an integer multiple of ␲, tan ␰ = 0, so 共unless g = 0兲
given by Eq. 共21兲, but now tan ␦ = −1; whenever ␰ is a half-integer multiple of ␲, tan ␰
␺k共⑀兲 = 0, 共32兲 → ± ⬁, so tan ␦ = 1. Clearly, ␦ does not approach a limit. The
case g = 0 is special, and it is best to treat it separately: For
and hence AH共2兲 共1兲
ig 共k⑀兲 + BHig 共k⑀兲 = 0, so 兩z兩 Ⰶ 1,
B H共2兲
ig 共k⑀兲 2i 2i
=− . 共33兲 H共1兲
0 共z兲 ⬇ 1 + 关ln共z/2兲 + C兴 = 1 + ln共␥z/2兲, 共40兲
A H共1兲
ig 共k⑀兲 ␲ ␲
Referring to Eqs. 共23兲 and 共24兲, we find that the phase shift where C = 0.577 215 is Euler’s constant and ␥ ⬅ eC
satisfies17 = 1.78 1072.20 Thus tan关arg H共1兲
0 共z兲兴 ⬇ 2 ln共␥z / 2兲 / ␲, and
B H共2兲
ig 共k⑀兲 ␲g 关H共1兲
ig 共k⑀兲兴
* hence
e2i␦ = i e␲g = − i 共1兲 e = − i 共1兲 . 共34兲
A Hig 共k⑀兲 Hig 共k⑀兲 ln共␥k⑀/2兲 + ␲/2
tan ␦ ⬇ . 共41兲
Notice that conservation of probability has been enforced by ln共␥k⑀/2兲 − ␲/2
Eq. 共32兲. Thus At this point we would like to send ⑀ → 0, to recover the
␲ pure 1 / x2 potential. Naively, Eq. 共31兲 suggests that E1 will
␦ = − arg关H共1兲
ig 共k⑀兲兴 − . 共35兲 go to −⬁—precisely the trap we were hoping to avoid. But
4 closer inspection reveals that the boundary condition, Eq.
This function is plotted in Fig. 6. 共27兲, only determines the product ␬1⑀. Suppose we stipulate
As the graphs suggest, arg关H共1兲 that E1 共and hence ␬1兲 remain constant as ⑀ → 0. From Fig. 5
ig 共x兲兴 is extremely steep near we see that this assumption forces g → 0, leaving16
the origin. Indeed, for 兩z兩 Ⰶ 1,18
1 + coth ␲g ⌫共1 + ig兲 2 −␲/g
H共1兲 ␬1 = e 共⑀ → 0,g → 0兲. 共42兲
ig 共z兲 ⬇ e
ig ln共z/2兲
− e−ig ln共z/2兲 , ␥⑀
⌫共1 + ig兲 ␲g
共36兲 The excited states, Eq. 共30兲, are squeezed out
19
so 2 −n␲/g
␬n = = ␬1e−共n−1兲␲/g → 0 共n = 2,3,4, . . . 兲, 共43兲
冉 冊
e
␲g ␥⑀
tan关arg H共1兲
ig 共z兲兴 ⬇ coth
2 but there remains a single bound state at finite 共though inde-
⫻tan关g ln共z/2兲 − arg ⌫共1 + ig兲兴, 共37兲 terminate兲 energy.
We can use Eq. 共42兲 to eliminate the cutoff ⑀ in favor of
and hence ␬1, in the scattering problem:

112 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 112

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 ␦共k兲 = tan−1 冋 ln共k/␬1兲 + ␲/2
ln共k/␬1兲 − ␲/2
. 册 共49兲

This scheme does not determine ␬1, and it forces ␣ → 1 / 4,

but the physical implications of the theory are perfectly sen-
sible.
The renormalization procedure we have just described
Fig. 7. The scattering phase shift ␦共k / ␬1兲, Eq. 共47兲, for g Ⰶ 1. The graph on may sound artificial, but it is not altogether unreasonable.22
the right shows the behavior near the origin 共the horizontal scale is in pow- After all, in practice we would have no way of knowing
ers of 10兲. whether the potential is really 1 / x2 all the way down to x
= 0; what we actually measure is the ground state energy and
the scattering phase shift. The former provides us with a
relation between ⑀ and g, Eq. 共42兲, but it does not determine
k⑀ =
k 2 −␲/g
e . 共44兲 either one separately, and the latter, Eq. 共49兲, stands as a
␬1 ␥ testable prediction irrespective of the actual 共but unmeasur-
able兲 value of the cutoff.23
So for low energy scattering, Eq. 共39兲 becomes What about the symmetry argument based on scale invari-
ance 共or, if you prefer, dimensional analysis兲 that seemed to
␰ ⬇ g ln共k/␬1兲 − ␲ , 共45兲 prove conclusively that the 1 / x2 potential can have no
ground state 共if it has one negative allowed energy, then
every negative energy is an eigenvalue兲? Well, we broke that
tan ␰ ⬇ tan关g ln共k/␬1兲兴 ⬇ g ln共k/␬1兲, 共46兲 symmetry when we introduced the cutoff, and the break per-
sists even as we eliminate ⑀ from the theory in favor of ␬1,
and Eq. 共38兲 reads and 共implicitly兲 send ⑀ → 0. This is an example of “anoma-
lous” symmetry breaking. There are three standard mecha-
ln共k/␬1兲 + ␲/2 nisms for breaking a symmetry in physics:
tan ␦ ⬇ . 共47兲
ln共k/␬1兲 − ␲/2 1. External 共or dynamical兲: An imposed force spoils the
symmetry 共for example, at the surface of the earth gravity
Curiously, the phase shift is independent of g 共although we breaks the three-dimensional isotropy of space兲.
have stipulated that g Ⰶ 1兲.21 For extremely low-energy scat- 2. Spontaneous: The ground state of a system is degenerate,
tering, k Ⰶ ␬1, the phase shift is evidently ␲ / 4 共see Fig. 7兲. and historical accident selects a particular one 共for ex-
Conclusion: The regularized potential, Eq. 共25兲, has a non- ample, the magnetization of a small piece of iron, which
problematic spectrum of discrete bound states, and a con- could have pointed in any direction, but in fact has to
tinuum of scattering states with well-defined phase shifts. point in some specific direction兲.
Naively, the limit ⑀ → 0 共which restores the pure 1 / x2 poten- 3. Anomalies: The process of renormalization breaks the
tial兲 reintroduces all of the pathological features of the origi- symmetry.24
nal. But closer examination reveals a loophole: If as we send
⑀ → 0 we simultaneously let g → 0, in such a way as to hold
␬1 constant 共Fig. 8兲, then a single bound state with energy IV. SELF-ADJOINT EXTENSIONS

ប2␬21 In Sec. II we examined the patient and diagnosed the ill-

E1 = − 共48兲 ness. In Sec. III we provided a partial cure. Now it is time to
2m identify the root cause of the disease and propose a more
comprehensive treatment. The fundamental problem with the
survives, and the scattering phase shift is given by 1 / x2 potential is that the Hamiltonian

H=−
ប2 d2
2m dx 2 −
a
x 2 = −
ប2 d2
2m dx
␣
2 + 2
x
冉 冊 共50兲

is not Hermitian 共more precisely, it is not self-adjoint兲. In

quantum mechanics self-adjoint operators occupy a privi-
leged position, because they alone represent observable
quantities. Most of our experience and intuition is predicated
on the self-adjointness of the Hamiltonian, and when this
fails, the intelligibility of the theory goes with it.
An operator is defined not only by its action, A, but also
by its domain, DA, the space of functions 兵␺其 on which it
acts. Physicists tend to forget the second part, because in
most cases the domain is not problematic. Of course, ␺ and
A␺ must lie in the Hilbert space of square-integrable func-
tions, L2 共in our case, on the interval 0 ⬍ x ⬍ ⬁兲, to ensure
that inner products are well-defined. In fact, to guarantee the
existence of the Hermitian conjugate 共or “adjoint”兲 A†,
Fig. 8. Graph of ⑀ as a function of g, Eq. 共42兲, with ␬1 = 0.1. 具␾兩A␺典 = 具A†␾兩␺典, 共51兲

113 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 113

Downloaded 02 Oct 2012 to 152.3.102.242. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
the allowed functions must be dense in L2.25 Typically there u±共x兲 = xs± , 共55a兲
will be other conditions on DA as well.
If there exists a domain such that for all ␺ and ␾ in DA with

具␾兩A␺典 = 具A␾兩␺典 共52兲 s± =

1
2
± 冑 1
4
− ␣. 共55b兲
共that is, the actions of A and A† are identical兲, then the op-
erator is Hermitian 共mathematicians would say “symmetric”兲 So, using ␺ = u+ and ␾ = u−, we have32
over that domain. However, it may happen that as long as ␺
is in DA 共the domain of the operator兲, ␾ can be in a larger
domain, DA† 共the domain of the adjoint兲,26 and yet Eq. 共52兲
冏冉 u−*
du+
dx
du*
− u+ −
dx
冊冏 0
* *
= x共s−兲 s+x共s+−1兲 − xs+s−*x共s−−1兲

= 共s+ − s−*兲x共s++s−−1兲 = 冑1 − 4␣ .
*
still holds. In that case 共DA† 傻 DA兲 we may be able to extend
the domain of the operator 共which will automatically restrict 共56兲
the domain of the adjoint兲 until the two domains coincide. In
this circumstance, with A = A† and DA† = DA, the operator is Unless ␣ = 1 / 4 共a special case that keeps recurring and to
said to be self-adjoint. For self-adjoint operators, in other which we shall return兲 some other condition must be im-
words, both the action and the domain of the adjoint are the posed.
same as for the operator itself. What if we insist that allowed functions vanish in a finite
The process we have sketched is called “self-adjoint ex- 共but arbitrarily small兲 neighborhood33 of the origin? Then the
tension.” You start with a Hermitian operator on a specified boundary term vanishes trivially, and H is Hermitian 共on this
domain, and extend DA 共thereby contracting DA†兲 until the domain兲. But if ␺ is in this very restricted domain, ␾ could
domains are identical. This process raises several questions, be any square-integrable function, and the boundary term
which were first addressed by Weyl and later generalized by will still vanish. So the domain of the adjoint is very much
von Neumann and Stone:27 How can you tell whether a given larger than the domain of the operator, and hence H is not
operator admits a self-adjoint extension? Is the extension 共if self-adjoint. Question: Does H admit a self-adjoint extension,
it exists兲 unique? How do you construct the self-adjoint do- and if so, what is the self-adjoint domain?34 In von Neu-
main? The answers are buried in abstruse mathematical lit- mann’s procedure the first step is to look for eigenfunctions
erature that is largely inaccessible to physicists,28 but two of H with imaginary eigenvalues:35
recent articles provide a relatively straightforward guide for
H␾± = ± i␩␾± , 共57兲
the uninitiated.29,30
We begin by asking whether H in Eq. 共50兲 is Hermitian. where ␩ is real and positive.36 Thus

冉 冊
Suppose ␾共x兲 and ␺共x兲 are two functions in L2共0 , ⬁兲 such
d2 ␣
that H␾ and H␺ are also in L2共0 , ⬁兲. Using integration by
2 + 2 ␾ ± = ␬ ±␾ ± , 共58兲
2

parts 共twice兲, we have dx x

冕 冉 冊
⬁ where
− ប2 d 2␺ ␣
具␾兩H␺典 =
2m
␾* + ␺ dx
dx2 x2
共53a兲 冑⫿i2m␩
0 ␬± ⬅ = e⫿i␲/4␤ , 共59兲
ប

=− 冋冏 冏 冏 冏
ប2
2m
␾*
d␺
dx
⬁
−␺
d␾*
dx
⬁
with ␤ ⬅ 冑2m␩ / ប. The general solution is
␾±共x兲 = 冑x关A±H共1兲 共2兲
ig 共i␬±x兲 + B±Hig 共i␬±x兲兴, 共60兲

冊 册
0 0

+ 冕冉
0
⬁
d␾
2

dx
*
␣ *
2 + 2 ␾ ␺ dx
x
共53b兲 where 共as always兲 g = 冑␣ − 1 / 4, and A± and B± are arbitrary
constants. But H共2兲
ig 共i␬±x兲 is not in L2 共and hence not in DH†兲,
because at large x 关Eq. 共22b兲兴37

=具H␾兩␺典 −冏冉 冊冏
ប2
2m
␾*
d␺
dx
−␺
d␾*
dx
⬁

0
. 共53c兲 H共2兲
ig 共i␬±x兲 ⬇ 冑 2
␲ i ␬ ±x
e−i共i␬±x−␲/4兲e−␲g/2

Evidently H is Hermitian,

具␾兩H␺典 = 具H␾兩␺典, 共54兲

= 冑 2 −␲g/2 ␬ x
␲ ␬ ±x
e e ±, 共61兲

if the boundary term in Eq. 共53c兲 is zero 共for all ␺ and ␾兲. and because ␬ = ␤共1 ⫿ i兲 / 冑2, H共2兲
ig diverges exponentially.
There is no problem at infinity, where the functions and their Thus
derivatives can safely be taken to vanish;31 the trouble arises ␾±共x兲 = A±冑xH共1兲
ig 共i␬±x兲. 共62兲
at the lower limit. If we want H to be Hermitian, we shall
have to restrict its domain. Earlier, we stipulated 共for reasons Let n+ be the number of linearly independent solutions for
of continuity兲 that wave functions go to zero at the origin, ␾+, and n− the corresponding number for ␾−; n1 and n2 are
but in spite of appearances this condition does not suffice to called “deficiency indices,” and they play a major role in the
kill the boundary term, because the derivatives can 共and in theory. Weyl and von Neumann showed that if n+ = n− = n,
the critical cases do兲 diverge. For example, we found in Eq. there exists an n2-parameter family of self-adjoint exten-
共15兲 that solutions to the Schrödinger equation behave near sions. 共If n = 0, the operator is already self-adjoint, and if
the origin like n+ ⫽ n−, there is no self-adjoint extension.兲 In our case, n+

114 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 114

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= n− = 1, and it remains only to characterize the one-
parameter family of self-adjoint domains. The prescription is
as follows: ␺ is in the self-adjoint domain if
x0 ⬅
1 G
␤ J
冉冊 i/2g
共74兲

is a free constant38—the anticipated parameter that charac-

具⌽兩H␺典 = 具H⌽兩␺典, 共63兲 terizes the particular self-adjoint extension. For example, if
where ⌽ ⬅ ␾+ + ␭␾− for some ␭, which is to say 关Eq. 共53c兲兴 ␣ = 0 共so g = i / 2兲, Eq. 共73兲 reduces to
if

lim ⌽*冋 d␺
−␺
d⌽ *
册 = 0. 共64兲
1
冑x 冋 x0
d␺
dx
册
+ ␺ → 0, 共75兲

x→0 dx dx which is the self-adjoint extension for the free particle on 0

Evidently we need to know the behavior of ␾± for small x. 艋 x ⬍ ⬁.39
From Eq. 共36兲, In the critical case g = 0 共␣ = 1 / 4兲 Eq. 共65兲 is replaced by

1 + coth ␲g ᐉ± 2i
H共1兲 ig ln共i␬±x/2兲 H共1兲
0 共i␬±x兲 ⬇ 1 − + ln共␥␤x/2兲 共76兲
ig 共i␬±x兲 ⬇ e 2 ␲
⌫共1 + ig兲
⌫共1 + ig兲 关see Eq. 共40兲兴, and

冋 册
− e−ig ln共i␬±x/2兲 , 共65兲
␲g
␾± ⬇ A±冑x ± + ln共␥␤x/2兲 ,
1 2i
共77兲
provided that g ⫽ 0. But ␬± = exp共⫿i␲ / 4兲␤, so 2 ␲

ln共i␬±x/2兲 = ln共␤x兲 − ln 2 + iᐉ±␲/4, 共66兲 so

where ᐉ+ ⬅ 1 and ᐉ− ⬅ 3. Thus ⌽ ⬇ 冑x关G + J ln共␥␤x/2兲兴, 共78兲

eig ln共i␬±x/2兲 = eig ln ␤xe−ig ln 2e−ᐉ±g␲/4 共67兲 with

and hence G ⬅ 21 共A+ − ␭A−兲, 共79a兲

␾+ ⬇ A+冑x关Deig ln ␤x − Fe−ig ln ␤x兴, 共68a兲 2i

J⬅ 共A+ + ␭A−兲, 共79b兲
␲
␾− ⬇ A−冑x关De −␲g/2 ig ln ␤x
e − Fe ␲g/2 −ig ln ␤x
e 兴, 共68b兲
and
where

冋 册
d⌽ 1
⬇ 关G + 2J + J ln共␥␤x/2兲兴. 共80兲
D ⬅ e−ig ln 2e−␲g/4
1 + coth ␲g
, 共69a兲 dx 2 冑x
⌫共1 + ig兲
According to Eq. 共64兲, then, a function ␺ is in the self-

F ⬅ eig ln 2e␲g/4 冋 ⌫共1 + ig兲

␲g
. 册 共69b兲
adjoint domain if

冑x ln共x/x0兲 d␺ − 1
冋 1+
1
册
ln共x/x0兲 ␺ → 0, 共81兲
dx 冑x 2
It follows that
where in this case x0 ⬅ 共2 / ␥␤兲e−G/J is the free parameter
⌽ ⬇ 冑x关Geig ln ␤x − Je−ig ln ␤x兴 共x Ⰶ 1兲, 共70兲
characterizing the particular self-adjoint extension.40
where Where does all this leave us? If we want the 1 / x2 Hamil-
tonian, Eq. 共50兲, to be self-adjoint, we must tighten up the
G ⬅ D共A+ + ␭A−e−␲g/2兲, 共71a兲 naive boundary condition ␺共0兲 = 0, Eq. 共4兲, in favor of a self-
adjoint extension 关Eq. 共73兲 if ␣ ⫽ 1 / 4 and Eq. 共81兲 for the
J ⬅ F共A+ + ␭A−e␲g/2兲. 共71b兲 critical case ␣ = 1 / 4兴.41 The choice of a particular extension
Therefore 共which is to say, a particular value of x0兲 is arbitrary, and in

冋冉 冊 冉 冊 册
this sense there exists an entire one-parameter family of dis-
d⌽ 1 1 1 tinct physical theories described by the 1 / x2 potential. Ques-
⬇ + ig Geig ln ␤x − − ig Je−ig ln ␤x .
dx 冑x 2 2 tion: Do they admit reasonable bound state spectra? We
know that the normalized eigenstates of H are given by Eq.
共72兲 共18兲:
Using Eqs. 共70兲 and 共72兲, the boundary condition 关Eq. i␲
共64兲, or more simply its complex conjugate兴 becomes ␺␬共x兲 = A冑xKig共␬x兲 = A冑xH共1兲
ig 共i␬x兲, 共82兲

冋冉 冊
2
冑x关e2ig ln共x/x0兲 − 1兴 d␺
*
where ␬ ⬅ 冑−2mE / ប. Which 共if any兲 of these reside in the
1 1
− + ig e2ig ln共x/x0兲
dx 冑x 2 self-adjoint domain of H?

− 冉 冊册
1
2
− ig ␺* → 0 共73兲
In the critical case g = 0, Eq. 共40兲 yields 共for small x兲
␺␬共x兲 ⬇ − A冑x ln共␥␬x/2兲, 共83兲
共as x → 0兲, where and

115 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 115

Downloaded 02 Oct 2012 to 152.3.102.242. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
 d␺␬
dx
⬇−A
1
冑x
1
冋
1 + ln共␥␬x/2兲 .
2
册 共84兲
ACKNOWLEDGMENTS

Barry Holstein introduced us to this problem; we thank

Inserting these expressions into Eq. 共81兲 yields the condition him and Horacio Camblong for interesting discussions.
A ln共␥␬x0 / 2兲 → 0. This holds only if ␥␬x0 / 2 = 1, which is to Thanks also to Darrell Schroeter for help with Fig. 6共b兲.
say, if ␬ = 2 / ␥x0. Evidently there is exactly one bound state
for ␣ = 1 / 4, just as we found 共by a completely different a兲
Electronic mail: essin@berkeley.edu
route兲 in Sec. III 关Eq. 共48兲兴. But we cannot calculate the b兲
Electronic mail: griffith@reed.edu
allowed energy, because it depends on the arbitrary param- 1
Of course, V共x兲 is unbounded as x → 0, but this by itself is not
eter x0 共just as, in the renormalization method, it depended problematic—the same is true of the Coulomb potential and the delta
on the arbitrary cutoff ⑀兲. function well. Our study of the 1 / x2 potential was inspired by 共and in the
The case g ⫽ 0 is a little more complicated. Equation 共28兲 early parts modeled on兲 S. A. Coon and B. R. Holstein, “Anomalies in
gives 共for small x兲 quantum mechanics: The 1 / r2 potential” Am. J. Phys. 70, 513–519
共2002兲.
␺␬共x兲 ⬇ B冑x sin ␪ ,
2
共85兲 Is there any other potential with this defect? In other words, is there
another function of position that contains no dimensional constants and
where B ⬅ −A冑␲ / g sinh共␲g兲 and has the same units as 1 / x2? There is—the two-dimensional delta function,
␦2共r兲 = ␦共x兲␦共y兲. Our story can be told using the two-dimensional delta
␪共x兲 ⬅ g ln共␬x/2兲 − arg ⌫共1 + ig兲. 共86兲 function, but because the extra dimension makes the details a little more
cumbersome and because the 1 / x2 potential is arguably more realistic, we
It follows that prefer to consider it as the model. In fact, for states with circular sym-
d␺␬
dx
⬇B
1 1
冋
冑x 2 sin ␪ + g cos ␪ . 册 共87兲
metry the two-dimensional delta function is mathematically equivalent to
the 1 / x2 potential with ␣ = 1 / 4 关Eq. 共3兲兴—see Exercise 1 in Ref. 30. For
discussion of the two-dimensional delta function see L. R. Mead, and J.
Godines, “An analytical example of renormalization in two-dimensional
According to Eq. 共73兲, then, ␺␬ is in the self-adjoint domain quantum mechanics,” Am. J. Phys. 59, 935–937 共1991兲; R. Jackiw,
if “Delta-function potentials in two- and three-dimensional quantum me-

冉 冊
chanics,” in M. A. B. Bég Memorial Volume, edited by A. Ali and P.
2 Hoodbhoy 共World Scientific, Singapore, 1991兲, pp. 25–42.
g ln + arg ⌫共1 + ig兲 = n␲ 共n = 0, ± 1, ± 2, . . . 兲 3
Notice that if ␺共x兲 is normalizable, so too is ␺␤共x兲.
␬x0 4
K. S. Gupta and S. G. Rajeev, “Renormalization in quantum mechanics,”
共88兲 5
Phys. Rev. D 48, 5940–5945 共1993兲.
G. Arfken and H.-J. Weber, Mathematical Methods for Physicists 共Aca-
for g real 共␣ ⬎ 1 / 4兲, or 2 / ␬x0 = 1 for g imaginary 共␣ ⬍ 1 / 4兲. demic, Orlando 共2000兲, 5th ed., Chap. 11. The other solution, 冑xIig共␬x兲,
In the former case we obtain an infinite set of eigenvalues; in 6
diverges for large x.
the latter case just one. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Prod-
ucts 共Academic, San Diego, 共1980兲, Eqs. 共6.576.4兲 and 共8.332.3兲. Inci-
Conclusion: To make the 1 / x2 Hamiltonian self-adjoint we
dentally, states with different ␬ are not orthogonal.
are obliged to impose more stringent boundary conditions 7
One might ask what the usual approximation schemes have to say about
关Eq. 共73兲 or 共81兲兴 than we naively supposed. This imposition this potential. Not much. For the general power law V共x兲 = ax␯ on 0 ⬍ x
necessarily introduces a free parameter x0 with the dimen- ⬍ ⬁, the WKB approximation yields En ⬇ a关共n − 1 / 4兲ប冑2␲ / ma⌫共1 / ␯
sions of length, thereby breaking the scale invariance of the + 3 / 2兲 / ⌫共1 / ␯ + 1兲兴2␯/共␯+2兲 关see D. J. Griffiths, Introduction to Quantum
theory. The result is a reasonable spectrum of allowed states, Mechanics 共Prentice Hall, Englewood Cliffs, NJ, 2004兲, 2nd ed., Problem
whose energies, however, we are unable to predict, because 8.11兴, and the exponent is infinite when ␯ = −2. The variational principle
only confirms what we already knew—that the ground state is lower than
they depend on the value of the arbitrary parameter.42 every negative energy.
8
Reference 6, p. 962, Eqs. 共3兲 and 共4兲.
V. CONCLUSION 9
Some authors use a plus sign in Eq. 共24兲, which adds ␲ / 2 to the phase
shift. We prefer the minus sign, because it reduces to ␦ = 0 when the
The 1 / x2 potential is clearly problematic. We can fix it potential is zero.
10
共sort of兲 by renormalization or by self-adjoint extension, but For a list of accessible references see C. V. Siclen, “The one-dimensional
a reasonable person would likely conclude that the problem hydrogen atom,” Am. J. Phys. 56, 9–10 共1988兲.
11
Indeed, because allowed energies must exceed Vmin, E1 ⬎ −a / ⑀2.
itself is artificial and unphysical—maybe there is no such 12
For these parameters ␬1⑀ = 1.024 645, ␬2⑀ = 0.350 972, ␬3⑀ = 0.122 830,
thing as a 1 / x2 potential in nature. Perhaps some potentials and ␬4⑀ = 0.043 089.
are just plain illegal in quantum mechanics. It seems odd, 13
For the ground state this inequality would appear to require g Ⰶ 3 共see
though, that we never encounter such difficulties in classical Fig. 5兲, but in practice the approximation is good up to g = 3. For the
mechanics. excited states ␬ is smaller, and the approximation is valid for even higher
Well, in the first place there are classical precursors.43 14
g.
Reference 5, Eqs. 共11.112兲 and 共11.118兲, and Ref. 6, Eqs. 共8.331兲 and
Moreover, there do exist systems represented 共at least, to
共8.332兲.
good approximation兲 by a 1 / x2 potential 共at any rate by its 15
This holds for g ⬍ 3, as we can easily confirm by comparing the graph of
three-dimensional analog兲. The best example is the motion of Eq. 共30兲 with Fig. 5. For larger values of g the approximation itself is
a charged particle in the field of a stationary electric dipole, invalid for the ground state. Incidentally, Eq. 共29兲 has solutions for nega-
for instance, an electron in the vicinity of a polar molecule. tive n, but these are spurious, because they violate the assumption ␬⑀
Here the potential is −ep cos ␪ / r2, and 共surprisingly兲 the ra- Ⰶ 1.
16
The limiting case g = 0 is obviously problematic—indeed, K0共z兲 has no
dial Schrödinger equation is mathematically identical to Eq.
zeros for positive z. For 兩z兩 Ⰶ 1, ⌫共1 + z兲 ⬇ 1 − Cz, where C = 0.577 215 is
共2兲.44 The critical parameter ␣ = 1 / 4 was noted in early stud- Euler’s constant, so for small g arg ⌫共1 + ig兲 ⬇ −Cg, and Eq. 共30兲 is re-
ies of this system,45 which has attracted renewed attention placed by ␬n = 共2 / ␥⑀兲exp共−n␲ / g兲, where ␥ ⬅ exp共C兲 = 1.781 072. See
recently.46 Like it or not, we have to take this problem seri- Ref. 6, Eq. 共8.321.1兲, and p. xxviii.
We used the identity H共2兲 共1兲
关H共1兲
17 −␲g
ously. ig 共x兲 = 关H−ig共x兲兴 = e ig 共x兲兴 共valid for real x
* *

116 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 116

Downloaded 02 Oct 2012 to 152.3.102.242. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
 and real g兲. See Ref. 6, p. 969. not go to zero at infinity, but there is no penalty for excluding them here.
We used Ref. 6, Eq. 共8.405.1兲, to express H共1兲
18
␯ in terms of J␯ and N␯, Eq.
See, for example, D. V. Widder, Advanced Calculus 共Dover, New York,
共8.440兲 to approximate J␯, Eq. 共8.443兲 to approximate N␯, and Eq. 1998兲, 2nd ed., p. 325.
共8.332.3兲 to eliminate ⌫共1 − ig兲.
32
We assume here that ␣ ⬍ 1 / 4; for ␣ ⬎ 1 / 4 we could run the same argu-
19
Reference 6, Eqs. 共8.331兲 and 共8.332.1兲. ment using ␺ = ␾ = u+ to obtain the same result.
20
Reference 6, Eqs. 共8.441.1兲 and 共8.444.1兲, and p. xxviii. 33
More explicitly, functions in DH vanish if 0 艋 x 艋 ⑀ or x 艌 ␶ for arbitrarily
21
Note that this is a correlated limit in which ⑀ and g both go to zero in small ⑀ and arbitrarily large ␶.
such a way as to hold ␬1 in Eq. 共42兲 fixed. This is not the same as going 34
We follow the treatment in Ref. 30, where the special case ␣ = 1 / 4 is
straight to g = 0 and then letting ⑀ → 0 关Eq. 共41兲兴, which only reproduces posed as an exercise.
the limiting value ␲ / 4. 35
Of course, the eigenvalues of a Hermitian operator are real, so ␾+ and ␾−
22
This is not the only way to tame the 1 / x2 potential. Other regularizations cannot be in DH; rather, the eigenfunctions we seek lie in DH†.
have been proposed. See, for example, C. Schwartz, “Almost singular 36
Mathematicians usually take ␩ = 1, but this choice offends the physicist’s
potentials,” J. Math. Phys. 17, 863–867 共1976兲; H. E. Camblong, L. N. concern for dimensional consistency. In any case, it combines with other
Epele, H. Fanchiotti, and C. A. Garca Canal, “Dimensional transmutation arbitrary constants at the end.
and dimensional regularization in quantum mechanics, I. General theory,” 37
This approximation assumes Re共ig兲 ⬎ −1 / 2, which is fine as long as ␣
and “II. Rotational invariance,” Ann. Phys. 287, 14–100 共2001兲. Our ⬎ 0 共the potential is attractive兲. It is of some interest to explore self-
approach follows Ref. 4. It is important in principle to demonstrate that adjoint extensions of the repulsive 1 / x2 potential, but we shall not do so
all regularizations lead to the same physical predictions. If they do not, here.
38
the theory is non-renormalizable and there is very little that can be done Here x0 is simply a convenient packaging of the arbitrary constants m, A±,
with it. ␭, and ␩. It is clear from Eq. 共73兲 关if not from Eq. 共74兲兴 that x0 carries the
23
Of course, if we could detect several bound states, or measure the phase dimensions of length, and hence the choice of a particular self-adjoint
shifts at sufficiently high energy, then we could map out any departures extension entails breaking the scale invariance that led to all the difficul-
from the 1 / x2 potential. The question is whether we can make any sense ties in Sec. II.
39
out of the pure 1 / x2 potential; renormalization offers a means for doing This case violates our assumption in Ref. 37, so it should be taken with a
so. By the way, something very similar happens in quantum electrody- grain of salt. See Ref. 30, Example 1, for a more rigorous analysis.
namics, where the theory, naively construed, yields an infinite mass for 40
This result agrees with Eq. 共80兲 of Ref. 30, with r → x and ␸ → ␺ / x.
41
the electron. The introduction of a cutoff renders the mass finite but The term “self-adjoint extension” is potentially misleading, because at
indeterminate. We take the observed mass of the electron as input and use first sight it appears to involve a contraction, not an expansion, of the
it to eliminate any explicit reference to the cutoff. The resulting renor- domain. The point is that you must start out with a Hermitian operator,
malized theory has been spectacularly successful, yielding by far the most and H is not Hermitian with respect to the set of functions that satisfy the
precise 共and precisely confirmed兲 predictions in all of physics. boundary condition ␺共0兲 = 0. That is why we first had to restrict the
24
The classic example of an anomaly is the decay of the neutral pion, ␲0 domain 共see Ref. 33兲, and the “extension” is with respect to that much
→ ␥ + ␥, which could not occur without the breaking of chiral symmetry. more limited domain.
25 42
See, for instance, E. Zeidler, Applied Functional Analysis: Applications By changing the boundary conditions it could be argued that we are
to Mathematical Physics 共Springer, New York, 1997兲, pp. 116–117. radically altering the physical system, albeit at a single point. The process
26
Note the logical structure here: We choose DA, but DA† is then is analogous 共in some cases identical兲 to adding a delta function to the
determined—it is the space of functions ␾ 共in L2兲 such that if ␺ is in DA potential, and it is hardly surprising that this changes the spectrum of
then Eq. 共52兲 holds. allowed states. But the question was whether there is anything we could
27
H. Weyl, “Über gewöhnliche Differentialgleichungen mit Singularitäten do to salvage the 1 / x2 potential, and if the remedy is necessarily radical,
und die zugehörigen Entwicklungen willkürlicher Funktionen,” Math. so be it.
Ann. 68, 220–269 共1910兲; J. von Neumann, “Allgemeine Eigenwerttheo- 43
C. Zhu and J. R. Klauder, “Classical symptoms of quantum illnesses,”
rie Hermitescher Funktionaloperatoren,” ibid. 102, 49–131 共1929兲; M. H. Am. J. Phys. 61, 605–611 共1993兲; N. A. Wheeler, “Relativistic classical
Stone, “On one-parameter unitary groups in Hilbert space,” Ann. Math. fields,” unpublished notes, Reed College, 1973, pp. 246–251.
33, 643–648 共1932兲. 44
J.-M. Lévy-Leblond, “Electron capture by polar molecules,” Phys. Rev.
28
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert 153, 1–4 共1967兲.
Space 共Dover, New York, 1993兲; M. Reed and B. Simon, Methods of 45
M. H. Mittleman and V. P. Myerscough, “Minimum moment required to
Modern Mathematical Physics 共Academic, London, 1975兲; G. Hellwig, bind a charged particle to an extended dipole,” Phys. Lett. 23, 545–546
Differential Operators of Mathematical Physics 共Addison-Wesley, Read- 共1966兲; W. B. Brown and R. E. Roberts, “On the critical binding of an
ing, MA, 1964兲. electron by an electric dipole,” J. Chem. Phys. 46, 2006–2007 共1966兲.
29 46
G. Bonneau, J. Faraut, and G. Valent, “Self-adjoint extensions of opera- C. Desfrançois, H. Abdoul-Carime, N. Khelifa, and J. P. Schermann,
tors and the teaching of quantum mechanics,” Am. J. Phys. 69, 322–331 “From 1 / r to 1 / r2 potentials: Electron exchange between Rydberg atoms
共2001兲. and polar molecules,” Phys. Rev. Lett. 73, 2436–2439 共1994兲; Coon and
30
V. S. Araujo, F. A. B. Coutinho, and J. F. Perez, “Operator domains and Holstein 共Ref. 1兲; many articles by H. E. Camblong and collaborators,
self-adjoint operators,” Am. J. Phys. 72, 203–213 共2004兲. especially “Quantum anomaly in molecular physics,” Phys. Rev. Lett. 87,
31
There exist pathological functions that are square-integrable and yet do 220402-1–4 共2001兲.

117 Am. J. Phys., Vol. 74, No. 2, February 2006 A. M. Essin and D. J. Griffiths 117

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