The solution to a problem in quantum mechanics is generally a linear superposition of states. The solutions for double well potentials epitomize this property, and go even further than this: they can often be described by an effective model whose low energy features can be described by two states—one in which the particle is on one side of the barrier, and a second where the particle is on the other side. Then the ground state remains a linear superposition of these two macroscopic-like states. In this paper, we illustrate that this property is achieved similarly with an attractive potential that separates two regions of space, as opposed to the traditionally repulsive one. In explaining how this comes about we revisit the concept of “orthogonalized plane waves,” first discussed in 1940 to understand electronic band structure in solids, along with the accompanying concept of a pseudopotential. We show how these ideas manifest themselves in a simple double well potential, whose “barrier” consists of a moat instead of the conventional wall.

1.
Any undergraduate textbook in Quantum Mechanics will have a discussion of state superposition. More often than not this will start with two-state systems, like the ammonia molecule, made famous in the discussion in
R. P.
Feynman
,
R. B.
Leighton
, and
M.
Sands
,
The Feynman Lectures on Physics, Volume III
(
Addison-Wesley
,
Reading, MA
,
1965
).
The one-dimensional double well potential is discussed extensively in
E.
Merzbacher
,
Quantum Mechanics
, 3rd ed. (
Wiley
,
Hoboken, NJ
,
1998
),
and more recently in
V.
Jelic
and
F.
Marsiglio
, “
The double-well potential in quantum mechanics: a simple, numerically exact formulation
,”
Eur. J. Phys.
33
,
1651
1666
(
2012
).
2.
In a discussion of some of this work with Don Page, he instantly coined the term, “quantum moat” that we use in this paper. We thank him for this, and for subsequent discussion concerning this problem.
3.
Although of course castle-builders always shored up the “moat-barrier” with a “wall-barrier.”
4.
In part, this notion was further explored in Pedro
L.
Garrido
,
Sheldon
Goldstein
,
Jani
Lukkarinen
, and
Roderich
Tumulka
, “
Paradoxical reflection in quantum mechanics
,”
Am. J. Phys.
79
,
1218
1231
(
2011
).
5.
Analytical solutions are possible also in the case of a finite-width square potential in an infinite square well (see, for example,
T.
Dauphinee
and
F.
Marsiglio
, “
Asymmetric wave functions from tiny perturbations
,”
Am. J. Phys.
83
,
861
866
(
2015
)), but there are further complications that makes analysis of this configuration rather complicated.
6.
F.
Bloch
, “
Über die quantenmechanik der elektronen in kristallgittern
,”
Z. Phys.
52
,
545
555
(
1929
),
in German. See also
F.
Bloch
, “
Memories of electrons in crystals
,”
Proc. R. Soc. Lond. A
371
,
24
27
(
1980
), where some context for this Theorem is provided by its originator.
7.
A good introductory text is
N. W.
Ashcroft
and
N. D.
Mermin
,
Solid State Physics
, 1st ed. (
Brooks/Cole
,
Belmont, CA
,
1976
).
8.
A good slightly more advanced text is
G.
Grosso
and
G. P.
Parravicini
,
Solid State Physics
(
Academic Press
,
Toronto
,
2000
).
9.
R. de L.
Kronig
and
W. G.
Penney
, “
Quantum mechanics of electrons in crystal lattices
,”
Proc. R. Soc. Lond. A
,
130
,
499
513
(
1931
).
10.
E. P.
Wigner
and
F.
Seitz
, “
On the constitution of metallic sodium
,”
Phys. Rev.
43
,
804
810
(
1933
);
E. P.
Wigner
and
F.
Seitz
On the constitution of metallic sodium. II
,”
Phys. Rev.
46
,
509
524
(
1934
).
11.
J. C.
Slater
, “
The electronic structure of metals
,”
Rev. Mod. Phys.
6
,
209
280
(
1934
);
J. C.
Slater
Wave functions in a periodic potential
,”
Phys. Rev.
51
,
846
851
(
1937
).
12.
C.
Herring
, “
A new method for calculating wave functions in crystals
,”
Phys. Rev.
57
,
1169
1177
(
1940
).
13.
J. C.
Phillips
and
L.
Kleinman
, “
New method for calculating wave functions in crystals and molecules
,”
Phys. Rev.
116
,
287
294
(
1959
).
14.
E.
Antončík
, “
Approximate formulation of the orthogonalized plane-wave method
,”
J. Phys. Chem. Solids
10
,
314
320
(
1959
).
15.
The derivation of this wave function follows closely the steps taken in section 2.5 of
D. J.
Griffiths
,
Introduction to Quantum Mechanics
, 2nd ed. (
Cambridge U.P.
,
Cambridge
,
2017
). The most significant deviations are the change in boundary conditions, where ψ(a/2)=ψ(a/2)=0replaceslimx±ψ(x)=0, and in the appearance of two independent solution sets for values of k that give the alternating even and odd solutions.
16.
In the present context “scattering states” refers to solutions with E > 0 while “bound states” refers to solutions with E < 0. The latter are present only when an attractive potential inside the infinite square well is used.
17.
F.
Marsiglio
, “
The harmonic oscillator in quantum mechanics: A third way
,”
Am. J. Phys.
77
,
253
258
(
2009
).
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