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.
References
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, “ The present situation in quantum mechanics: A translation of Schrödinger's ‘cat paradox’ paper
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. 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
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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
); and5.
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.
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, and M. A.
Dubson
, Modern Physics for Scientists and Engineers
, 2nd ed. (Prentice Hall
, Upper Saddle River, NJ
, 2004
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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
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28
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(1998
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G.
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D.
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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
), andB.
Thaller
, Advanced Visual Quantum Mechanics
(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
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, 2nd ed. (Pearson Prentice Hall
, Upper Saddle River, NJ
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);J. S.
Townsend
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, 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
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(1982
).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.
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For more sophisticated analyses of entanglement in one-dimensional scattering interactions, see
C. K.
Law
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70
(6), 062311-1
–4
(2004
);N. L.
Harshman
and P.
Singh
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41
(15), 155304
(2008
); andH. S.
Rag
and J.
Gea-Banacloche
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83
(4), 305
–312
(2015
).20.
When the potential for a two-particle system depends only on the separation distance (here ), 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
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85
, 698
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(2017
).22.
See, for example,
J. J. V.
Maestri
, R. H.
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, and M. J.
Páez
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(2000
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R.
Grobe
, K.
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, and J. H.
Eberly
, “ Measure of electron-electron correlation in atomic physics
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27
(16), L503
–L508
(1994
). See also the first two papers listed in Ref. 19.24.
For a more sophisticated treatment of hydrogen, see
P.
Tommasini
, E.
Timmermans
, and A. F. R.
de Toledo Piza
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66
(10), 881
–886
(1998
).25.
F. W.
Strauch
, “ Resource Letter QI-1: Quantum Information
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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
), and29.
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.
“Google Books Ngram Viewer,” <https://books.google.com/ngrams>.
31.
H. J.
Groenewold
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H.
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(1963
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J. L.
Park
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36
(3), 211
–226
(1968
).34.
For example,
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2
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(1972
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(1976
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–126
(1980
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Bell
, Speakable and Unspeakable in Quantum Mechanics
(Cambridge U. P.
, Cambridge
, 1987
), pp. 105
–110
.36.
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(1982
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and A.
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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
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,” Br. J. Philos. Sci.
35
(1), 25
–45
(1984
).43.
Jørlunde Film Denmark
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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.
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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;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.
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