Quantum entanglement occurs not just in discrete systems such as spins, but also in the spatial wave functions of systems with more than one degree of freedom. It is easy to introduce students to entangled wave functions at an early stage, in any course that discusses wave functions. Doing so not only prepares students to learn about Bell's theorem and quantum information science, but can also provide a deeper understanding of the principles of quantum mechanics and help fight against some common misconceptions. Here I introduce several pictorial examples of entangled wave functions that depend on just two spatial variables. I also show how such wave functions can arise dynamically, and describe how to quantify their entanglement.

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E.
Schrödinger
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Discussion of probability relations between separated systems
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E.
Schrödinger
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Die gegenwärtige situation in der quantenmechanik
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,
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812
,
E.
Schrödinger
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Die gegenwärtige situation in der quantenmechanik
,”
823
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(
1935
).
Translation by
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, “
The present situation in quantum mechanics: A translation of Schrödinger's ‘cat paradox’ paper
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Proc. Am. Philos. Soc.
124
,
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338
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Reprinted in
Quantum Theory and Measurement
, edited by
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Wheeler
and
W. H.
Zurek
(
Princeton U. P.
,
Princeton
,
1983
), pp.
152
167
. Schrödinger's term for entanglement in the original German was Verschränkung.
3.
Schrödinger was responding to the newly published EPR paper, which used the concept of entangled states (though not the word “entangled”) to argue that quantum mechanics is incomplete:
A.
Einstein
,
B.
Podolsky
, and
N.
Rosen
, “
Can quantum-mechanical description of physical reality be considered complete?
,”
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47
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(
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4.
As discussed in the appendix, it appears that no quantum mechanics textbook used the word “entanglement” until 1998. Some more recent textbooks that don't use the word at all include
R. W.
Robinett
,
Quantum Mechanics
, 2nd ed. (
Oxford U. P.
,
Oxford
,
2006
); and
B. C.
Reed
,
Quantum Mechanics
(
Jones and Bartlett
,
Sudbury, MA
,
2008
).
5.
See supplementary material at http://dx.doi.org/10.1119/1.5003808 for answers to the exercises, a tutorial on plotting two-dimensional wave functions with Mathematica (also at <http://physics.weber.edu/schroeder/quantum/plottutorial.pdf>), and software for calculating wave functions such as those shown in Figs. 5 and 6 (also at <http://physics.weber.edu/schroeder/software/EntanglementInBox.html> and <http://physics.weber.edu/schroeder/software/CollidingPackets.html>).
6.
See, e.g.,
J. R.
Taylor
,
C. D.
Zafiratos
, and
M. A.
Dubson
,
Modern Physics for Scientists and Engineers
, 2nd ed. (
Prentice Hall
,
,
2004
), Sec. 8.3;
A.
Goswami
,
Quantum Mechanics
, 2nd ed. (
Wm. C. Brown
,
Dubuque, IA
,
1997
), Sec. 9.2.
7.
There does exist at least one textbook that has a nice treatment of superposition states for the two-dimensional infinite square well:
D.
Park
,
Introduction to the Quantum Theory
, 3rd edition (
McGraw-Hill
,
New York
,
1992
), Sec. 6.1 (Dover reprint, 2005). Notably, Park's illustrations are mere line drawings showing the wave function node locations—presumably because his earlier editions predate today's computer graphics tools. Now that those tools are widely available, there is one less barrier to visualizing two-dimensional wave functions.
8.
Many authors have recognized the important distinction between entanglement of two (or more) particles and entanglement of different degrees of freedom of a single particle. For example, only the former enables EPR-Bell nonlocality demonstrations. See
R. J. C.
Spreeuw
, “
A classical analogy of entanglement
,”
Found. Phys.
28
(3),
361
374
(
1998
), in which the author suggests the terms nonlocal entanglement for the multi-particle case and classical entanglement for the single-particle case.
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G.
Zhu
and
C.
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D.
Styer
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Common misconceptions regarding quantum mechanics
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Am. J. Phys.
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31
43
(
1996
), item 1.
11.

The term “cat state” refers to Schrödinger's cat (see Ref. 2), which is supposedly in a superposition of alive and dead states. Nowadays many physicists use this term to describe any quantum state that is best thought of as a superposition of two other states, especially when the difference between those two states is large or important. In the example used here the two states are localized around points that are well separated from each other, so in a sense the particle is in two places at once.

12.
For a detailed discussion of representing complex phases as color hues, see, for example,
B.
Thaller
,
Visual Quantum Mechanics
(
Springer
,
New York
,
2000
), and
B.
Thaller
,
(
Springer
,
New York
,
2005
). See the electronic supplement to this article, Ref. 5, for a tutorial on plotting wave functions using Mathematica.
13.
For example,
D. J.
Griffiths
,
Introduction to Quantum Mechanics
, 2nd ed. (
Pearson Prentice Hall
,
,
2005
);
D. H.
McIntyre
,
(
Pearson
,
Boston
,
2012
);
J. S.
Townsend
,
A Modern Approach to Quantum Mechanics
, 2nd ed. (
University Science Books
,
Mill Valley, CA
,
2012
).
14.

Misconception 3 is closely related to the misconception that the wave function is a function of “regular three-dimensional position space” rather than configuration space, as described by Styer, Ref. 10, item 3.

15.

Few quantum mechanics textbooks contain even a single plot of a two-particle wave function. An exception is McIntyre, Ref. 13, pp. 418–419.

16.
This exercise was inspired by
D.
Styer
, Notes on the Physics of Quantum Mechanics,
2011
, p.
150
, <http://www.oberlin.edu/physics/dstyer/QM/PhysicsQM.pdf>.
The point was made in more generality by
R. P.
Feynman
, “
Simulating physics with computers
,”
Int. J. Theor. Phys.
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,
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488
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17.

Although it is interactions that cause unentangled particles to become entangled, not every entangled state must result from an interaction. For example, a system of two identical fermions is inherently entangled, although the indistinguishability of the particles and the presence of spin introduce further subtleties.

18.
This system has been discussed previously by
J. S.
Bolemon
and
D. J.
Etzold
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,”
Am. J. Phys.
42
,
33
42
(
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);
J. R.
Mohallem
and
L. M.
Oliveira
, “
Correlated wavefunction of two particles in an infinite well with a delta repulsion
,”
Am. J. Phys.
58
(6),
590
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(
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);
J.
Yang
and
V.
Zelevinsky
, “
Short-range repulsion and symmetry of two-body wave functions
,”
Am. J. Phys.
66
(3),
247
251
(
1998
); and
E. A.
Salter
,
G. W.
Trucks
, and
D. S.
Cyphert
, “
Two charged particles in a one-dimensional well
,”
Am. J. Phys.
69
(2),
120
124
(
2001
).
19.
For more sophisticated analyses of entanglement in one-dimensional scattering interactions, see
C. K.
Law
, “
Entanglement production in colliding wave packets
,”
Phys. Rev. A
70
(6),
062311-1
4
(
2004
);
N. L.
Harshman
and
P.
Singh
, “
Entanglement mechanisms in one-dimensional potential scattering
,”
J. Phys. A
41
(15),
155304
(
2008
); and
H. S.
Rag
and
J.
Gea-Banacloche
, “
Wavefunction exchange and entanglement in one-dimensional collisions
,”
Am. J. Phys.
83
(4),
305
312
(
2015
).
20.
When the potential for a two-particle system depends only on the separation distance (here $x 2 − x 1$), the time-dependent Schrödinger equation is separable in terms of the center-of-mass and relative coordinates. See, for example,
D. S.
Saxon
,
Elementary Quantum Mechanics
(
Holden-Day
,
San Francisco
,
1968
), Sec. VIII 2 (Dover reprint, 2012). Whether this separation is physically useful depends on how the initial state is prepared and on what measurements on the final state one might perform. Also note that the potential used in Fig. 5 is not separable in this way, due to the presence of the external square-well trap.
21.
D. V.
Schroeder
, “
The variational-relaxation algorithm for finding quantum bound states
,”
Am. J. Phys.
85
,
698
704
(
2017
).
22.
See, for example,
J. J. V.
Maestri
,
R. H.
Landau
, and
M. J.
Páez
, “
Two-particle Schrödinger equation animations of wave packet-wave packet scattering
,”
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68
(12),
1113
1119
(
2000
).
23.
R.
Grobe
,
K.
Rzażewski
, and
J. H.
Eberly
, “
Measure of electron-electron correlation in atomic physics
,”
J. Phys. B
27
(16),
L503
L508
(
1994
24.
For a more sophisticated treatment of hydrogen, see
P.
Tommasini
,
E.
Timmermans
, and
A. F. R.
de Toledo Piza
, “
The hydrogen atom as an entangled electron-proton system
,”
Am. J. Phys.
66
(10),
881
886
(
1998
).
25.
F. W.
Strauch
, “
Resource Letter QI-1: Quantum Information
,”
Am. J. Phys.
84
(7),
495
507
(
2016
).
26.
The idea for these plots comes from
D.
Styer
, “Visualization of Quantal Entangled States,” <http://www.oberlin.edu/physics/dstyer/TeachQM/entangled.pdf>.
27.
Similarly, one can say that a Stern-Gerlach device “entangles” the spin state of a particle with its spatial wave function. See, for example, Griffiths, Ref. 13, Eq. (4.173), and
E.
Merzbacher
,
Quantum Mechanics
,
3rd ed
. (
Wiley
,
New York
,
1998
), p.
406
.
28.
Two insightful accounts of the history of quantum mechanics during this time period are
L.
Gilder
,
The Age of Entanglement
(
Knopf
,
New York
,
2008
), and
D.
Kaiser
,
How the Hippies Saved Physics
(
Norton
,
New York
,
2011
).
29.

This summary is inevitably incomplete, and I would welcome the communication of additions and corrections from readers who are knowledgable about the history of the term “entanglement.”

30.
31.
H. J.
Groenewold
, “
On the principles of elementary quantum mechanics
,”
Physica
12
(7),
405
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(
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32.
H.
Margenau
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Measurements and quantum states: Part II
,”
Philos. Sci.
30
(2),
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33.
J. L.
Park
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Nature of quantum states
,”
Am. J. Phys.
36
(3),
211
226
(
1968
).
34.
For example,
I.
Fujiwara
, “
Quantum theory of state reduction and measurement
,”
Found. Phys.
2
,
83
110
(
1972
);
N.
Maxwell
, “
Toward a microrealistic version of quantum mechanics. Part II
,”
Found. Phys.
6
(6),
661
676
(
1976
).
35.
J. S.
Bell
, “
,”
9
,
121
126
(
1980
);
reprinted in
J. S.
Bell
,
Speakable and Unspeakable in Quantum Mechanics
(
Cambridge U. P.
,
Cambridge
,
1987
), pp.
105
110
.
36.
J. S.
Bell
, “
Bertlmann's socks and the nature of reality
,”
J. Phys. Colloq.
42
(C2),
41
62
(
1981
). Reprinted in Ref. 35, pp. 138–158.
37.
A.
Peres
and
W. H.
Zurek
, “
Is quantum theory universally valid?
,”
Am. J. Phys.
50
(9),
807
810
(
1982
). The quote from Ref. 36 in this paper contains the only use of “entangled” that I can find in AJP from between 1968 and 1990.
38.
J. F.
Clauser
and
A.
Shimony
, “
Bell's theorem: Experimental tests and implications
,”
Rep. Prog. Phys.
41
(12),
1881
1927
(
1978
).
39.
B.
d'Espagnat
, “
Nonseparability and the tentative descriptions of reality
,”
Phys. Rep.
110
,
201
264
(
1984
).
40.
L. E.
Ballentine
, “
Resource letter IQM-2: Foundations of quantum mechanics since the Bell inequalities
,”
Am. J. Phys.
55
(9),
785
792
(
1987
).
41.
P. C. W.
Davies
and
J. R.
Brown
,
The Ghost in the Atom
(
Cambridge U. P.
,
Cambridge
,
1986
). The introductory chapter of this book does speak of a quantum system being “entangled” with the macroscopic experimental apparatus (p. 12) and with our knowledge of the system (p. 34).
42.
A.
Shimony
, “
Contextual hidden variables theories and Bell's inequalities
,”
Br. J. Philos. Sci.
35
(1),
25
45
(
1984
).
43.
Jørlunde Film Denmark
, “Atomic Physics and Reality,” television documentary, 1985. Posted by YouTube user Muon Ray at <https://www.youtube.com/watch?v=BFvJOZ51tmc>. This video also includes interviews with Aspect, Bell, Bohm, and Wheeler—yet only Shimony uses the word “entanglement” in the included footage.
44.
N.
Herbert
,
Quantum Reality: Beyond the New Physics
(
Anchor Books
,
New York
,
1985
).
45.
J. S.
Bell
, “
Are there quantum jumps?
,” in
Schrödinger: Centenary Celebration of a Polymath
, edited by
C. W.
Kilmister
(
Cambridge U. P.
,
Cambridge
,
1987
), pp.
41
52
; reprinted in Ref. 35, pp. 201–212.
46.
G. C.
Ghirardi
,
A.
Rimini
, and
T.
Weber
, “
Unified dynamics for microscopic and macroscopic systems
,”
Phys. Rev. D
34
(2),
470
491
(
1986
).
47.
G. C.
Ghirardi
,
A.
Rimini
, and
T.
Weber
, “
Disentanglement of quantum wave functions: Answer to “Comment on ‘Unified dynamics for microscopic and macroscopic systems’”
,”
Phys. Rev. D
36
(10),
3287
3289
(
1987
).
48.
M. A.
Horne
,
A.
Shimony
, and
A.
Zeilinger
, “
Two-particle interferometry
,”
Phys. Rev. Lett.
62
,
2209
2212
(
1989
).
49.
A.
Shimony
, “
The reality of the quantum world
,”
Sci. Am.
258
(1),
46
53
(
1988
).
50.
A.
Shimony
, “
Conceptual foundations of quantum mechanics
,” in
The New Physics
, edited by
P.
Davies
(
Cambridge U. P.
,
Cambridge
,
1989
), pp.
373
395
.
51.
D. M.
Greenberger
,
M. A.
Horne
,
A.
Shimony
, and
A.
Zeilinger
, “
Bell's theorem without inequalities
,”
Am. J. Phys.
58
(12),
1131
1143
(
1990
).
52.
R.
Penrose
,
The Emperor's New Mind
(
Oxford
U. P., Oxford
,
1989
), pp.
269
, 297;
R.
Penrose
,
(
Oxford U. P.
,
Oxford
,
1994
), Sec. 5.17.
53.
E.
Merzbacher
, “
An evolving physical system: The state of the American Physical Society
,”
Phys. Today
44
(
8
),
42
46
(
1991
).
54.

Merzbacher, Ref. 27, p. 362.

55.
S.
Gasiorowicz
,
Quantum Physics
, 3rd ed. (
Wiley
,
Hoboken, NJ
,
2003
).