Juego Mecanica Cuantica

Using Quantum Games To Teach Quantum Mechanics, Part 1Ross D. Hoehn,*, Nick Mack, and Sabre Kais,

Department of Chemistry and Department of Physics, Purdue University, West Lafayette, Indiana 47907, United States

*S Supporting Information

ABSTRACT: The learning of quantum mechanics is contingentupon an understanding of the physical signicance of the mathematicsthat one must perform. Concepts such as normalization, super-position, interference, probability amplitude, and entanglement canprove challenging for the beginning student. Several class activitiesthat use a nonclassical version of tic-tac-toe are described to introduceseveral topics in an undergraduate quantum mechanics course.Quantum tic-tac-toe (QTTT) is a quantum analogue of classical tic-tac-toe (CTTT) and can be used to demonstrate the use ofsuperposition in movement, qualitative (and later quantitative)displays of entanglement, and state collapse due to observation.QTTT can be used to aid student understanding in several othertopics with the aid of proper discussion.

KEYWORDS: First-Year Undergraduate/General, High School/Introductory Chemistry, Upper-Division Undergraduate,Physical Chemistry, Humor/Puzzles/Games, Hands-On Learning/Manipulatives, Quantum Chemistry, Student-Centered Learning

Classical mechanics, whose approach was developed basedon Newtons new mathematics, was contemporaneouslyformulated alongside calculus. Both topics moved fromacademic investigation into undergraduate lecture halls, andin the case of Newtonian mechanics, earlier still, with itsconcepts being introduced prior to high school. Quantummechanics, developed in the 20th century, was required toadequately describe such experimental phenomena as black-body radiation, the photoelectric eect, and the atomicspectrum of hydrogen. The development of quantummechanics has led to description of phenomena such as thesuperposition principle, the ability of an unobserved quantumobject to exist in a superposition of multiple statessimultaneously; entanglement, spooky action at a distancewhere the state of one system aects that of another without adirect observable relationship connecting them; and interfer-ence, as matter exists in both particle and waveform withinquantum theory, matter interactions present wave phenomenonsuch as diraction and the properties of constructive anddestructive matterwave addition. Just as a rudimentaryunderstanding, at minimum, of classical mechanics becamenecessary for so many elds, an introduction into the conceptsof quantum mechanics is of growing importance.A students rst excursion into quantum mechanics can be

both overwhelming and daunting, even to an upper-divisionscience student. Understanding such concepts as wavefunctions, overlap integrals, and probability amplitudes arevital in mastering the subsequent material within the course. Atypical rst semester course in quantum mechanics focuses onthe Schrodinger picture and equation.13 Herein we presentseveral activities using quantum tic-tac-toe (QTTT), which is aquantum analogue of classical tic-tac-toe (CTTT), presented by

Allen Go,46 as a means of introducing and enforcing earlytopics in an introductory quantum mechanics course. Theactivities allow for introduction and discussion of probabilityamplitude, probability density, normalization, overlap, the innerproduct, and separability of states. It is the belief of the authorsthat QTTT can be used as an approachable, fun, and intuitivemeans of introducing these topics. It is the hope of the authorsthat this tool could act as a companion throughout instruction;after the students have been taught the game, the instructor canuse it as a stepping-stone to new topics and as an avenue forintuitive activities.The activities described, as well as other similar activities,

have been used to assist the understanding of various audiencesin anything from a brief understanding of concepts necessary toquantum computing to furthering a students understanding oftopics in their quantum mechanics classroom. The bulk of thematerial was used as assignments and Supporting Informationin an undergraduate quantum mechanics classroom to greatavail with students who did not grasp some early conceptswithin the course.

PHYSICAL CONCEPTS AND GAME PLAYBoard

The tic-tac-toe board is square and is divided into nine squaresubspaces. These subspaces will be referred to as principalsquares and will each carry a number to denote the particularsquare being referenced. The numbering pattern of theprincipal squares on the board is shown in Figure 1. Prior todiscussing the game play, some vocabulary and concepts are

Published: January 29, 2014

Activity

pubs.acs.org/jchemeduc

2014 American Chemical Society andDivision of Chemical Education, Inc. 417 dx.doi.org/10.1021/ed400385k | J. Chem. Educ. 2014, 91, 417422

introduced. The following four elements are underlyingphysical concepts that are necessary for game play and thustheir use is weaved within the description of the game.Spooky Marker. Named after Einsteins reference to

entanglement and hidden variable interactions as spookyaction at a distance.7 This is a direct consequence to thesystem being completely described through a nite number ofbasis functions of an observable. A coupled pair of electronsexist within a 0-spin state; that is to say that the wave functionof the pair is of the form: = 1/2(| + |). If oneobserves the state of a single electron within the pair, say it is inthe up-state, that observer incidentally knows the state of theother spin within the pair. Like CTTT markers, the spookymarker represents a single move of a single player during oneturn, yet a spooky marker exists within two separate principalsquares simultaneously.Superposition. As players have placed a pair of spooky

markers that represents their move for that turn, this move canbe said to exist as a superposition of the states (boardpositions) in which it may be realized. If player one, referred toas Alice, places spooky markers for her rst move into squares 1and 5, then the state of that move is the superposition of thetwo states: square 1 and square 5. All player moves withinQTTT are superposition moves. A typical teaching example ofthis is the superposition of spins separated through observationin the SternGerlach810 experiments, which are typicallydiscussed in introductory quantum mechanics courses. Afurther example, to which the students may have alreadybeen exposed, is the superposition of ammonia states bytunneling; the students may have discussed this already in theirorganic chemistry course with reference to nitrogen inver-sions11 and the topic can be expounded through a discussion ofthe MASER problem.12

Cyclic Entanglement. Entanglement is the correlationbetween parts of a system, induced through an interaction andmaintained in separation, which is independent of factors suchas position and momentum.13 In QTTT this would consist of agroup of spooky markers whose board positions are all self-referencing; as an example, Alices rst move (X1) exists in bothsquares 1 and square 5; the second players, referred to as Bob,rst move (O2) exists both within square 5 and square 7; andAlices second move (X3) is within both square 7 and square 1.In this way, the possible states of these moves are dependentupon each other in a similar fashion as to the spin states ofpaired electrons. The cyclic reference here is that X1 sharesprincipal square 1 with X3, X3 shares principal square 7 with O2,and nally O2 shares principal square 5 with X1. This series of

moves is shown in Figure 2 and will be made clearer in asample game.

State Collapse. A quantum system may exist in asuperposition of several states. Only one subordinate state isobserved when the state of the system is measured. An exampleof this would be a doublet spin system; the state of the singleelectron would be a superposition of up-spin and down-spin,yet when observed, a single electron will present only either anup-spin or a down-spin state. When a state collapse occursthrough observation within the game, spooky markers collapseinto CTTT marks.General Structure of the Game

The general structure of the game is similar to that of CTTT.The few caveats and expansions to the rules can be most easilyeshed-out through an example game. Game play begins asAlice places her rst pair of spooky markers on the board; anysuch move within the game will be denoted by |i

j, where represents the player to whom the marker belongs and will thustake on the values X or O, i denotes the turn when this markerwas placed, and j is the location on the board where the markerwas placed. She places her markers in principal squares 1 and 5.This means that her rst move, |1

X, is a super position withthe form: |1

X = 1/2(|1X1 + |1X5). Let us now have Bobplace his markers in principal squares 5 and 7; unlike theclassical tic-tac-toe game, the placement of a spooky marker inQTTT does not prevent either player from placing subsequentmarkers in a particular square. Alice retorts with markers inprincipal squares 7 and 1. With this last move, our game boardis now consistent with that in Figure 2. It can now be seen thatthe state of each of the spooky markers is a linear combinationof the two squares that it occupies and each position within thislinear combination is a position within a linear combinationdescribing another spooky marker. In Figure 2 it can now beseen that we have generated a cyclic entanglement betweenmarkers placed for 1

X, 2O and 3

X through their possible states(squares 1, 5, and 7).As a cyclic entanglement has been generated, it is time for a

player to make an observation on the system that will cause astate collapse of our spooky markers into classical markers. AsAlices last move was that which sealed the cyclic entanglement,it will be Bobs right to decide in which way the states willcollapse; this reciprocation of closure and observation wasdeveloped in hope to generate a fair game, although it was anad hoc rule implemented for the sake of fair game play (a more

Figure 1. The layout of the game board for either classical or quantumtic-tac-toe. This gure also displays the enumeration scheme that isused throughout this paper.

Figure 2. The eect of measurement on the system of cyclicentanglement can yield, at minimum, a pair of classical statescorresponding to the state in which X3 was observed.

Journal of Chemical Education Activity

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quantum mechanically accurate rule would be ipping a coin todecide the collapse). When an observation is made on thesystem, the states of the markers involved with the cyclicentanglement will collapse. Unlike a spooky marker, when aclassical marker occupies a board position, no other marker(neither classical nor quantum) may occupy that position.The two possible pathways that an observation could take are

also shown in Figure 2. We will rst state completely the logicof the upper path and then that of the lower path. If Bobchooses that Alices most recent move, |3

X = 1/2(|3X1 +|3

X7), should be observed in square 7, this would imply thatthe only state that |2

O = 1/2(|2O5 + |2O7) could take isthat of square 5 and thus the only state |1

X = 1/2(|1X1 +|1

X5) can manifest is that of square 1; all this due to the factthat this observation turns these spooky markers into classicalmarkers and thus exclusively occupy their observed site.If Bob had chosen the other path, 3

X would collapse insquare 1 forcing 1

X in square 1 and nally 2O in square 5. The

lower board is that which would occur if Bob chose to observe3X in square 1. It should also be noted that if a situation arises

consistent with Figure 3, there exist a pair (or more) or spooky

markers that are entangled with the cycle without both of itsstates being enveloped by the cycle. In these cases, theobservation will also eect a collapse upon the danglingmarker; the subsequent collapse of dangling markers can alsobe seen in Figure 3.Game play will continue in this manner until one of the

players has generated a three-in-a-row consisting of onlyclassical markers. It is possible that two players willsimultaneously win the game through the same observation.When this occurs, the player with the most recent SpookyMakers generating one of their winning classical markers loses;which is, in a way, to say rst in, rst out. When playing avolley of games, Go46 does propose that the winning playerduring a simultaneous victory be awarded 1 point and the loser1/2 point.

ACTIVITYWe present the following work as an instructor-guided inquiryactivity14,15 with the students divided into groups of two. Wepresent specic board examples as a means of discussion andinstructional guidance examples, as introductory courses havebeen shown to benet from strong instructor guidance.16 Amore natural experience would be allowing the students(postinstruction on the rules and teaching a specic

phenomenon) to play the game and come across thesephenomena on their own in an inductive learning style similarto a lab exercise.1720 QTTT could also be used as acontinuing-themed homework exercise as it can be used toexemplify many of the introductory topics in quantummechanics.Introducing the game rules and running a small example

game can take up to 15 min, whereas the average time to play asingle game is roughly 4 min. In the experience of theseauthors, the use of quantum tic-tac-toe lowers the level of fearassociated with introducing these early concepts, as it bothbuilds student condence and gives them a foothold on thematerial through a familiar mechanism. Students took to thegame enthusiastically and divorced of the quantum mechanicalconcepts, learning the game rules comes quickly. The mostdicult part in learning the game is recognizing the closedloops; it is suggested that the instructor select a student to actas a representative for all the students as the class plays againstthe instructor for a game; this method seems to reveal thepresent thought processes of the students, which can benetinstruction. These authors also found that the notions to bediscussed within the following sections of this paper benettedfrom introduction through QTTT as they are, at times, earlysigns of student understanding of quantum mechanics.We will maintain the use of Alice as player X and Bob as

player O, which is appropriate as the groups are of two players.Player names, Alice and Bob, were purposefully chosen, asdiscussion of pairs entangled particles uses the notationparticles A and B; from this notation, observers at each endof the system are often referred to as Alice (for A) and Bob (forB).13

PROBABILITY AMPLITUDE, SIGN SYMMETRY, ANDPROBABILITY DENSITY

The fundamental quantity within the Schrodinger picture ofquantum mechanics is the wave function, (x). (x) are thesolutions to the second-order dierential wave equationdescribing the total system energy of a particle.21 The use ofeither QTTT or CTTT does not lend itself to the introductionof the Schrodinger equation as there are no intuitive norappropriate methods for the student to connect game play toenergy. Yet use of QTTT has proven benecial in theexplanation and discussion of several properties of the wavefunction, especially topics such as normalization and signsymmetry of the probability amplitude.Wave functions, as stated by the rst postulate of quantum

mechanics,1 show how the state of their system evolves in time.The use of Gaussian-type functions in the description of moveslends itself immediately as a means of emphasizing the signinvariance of the probability density. We will begin by dening:

= + g x y( , ) ejx y[( ) ( ) ]/2j

xjy2 2 2

(1)

where j denotes the board space in which the Gaussian functionresides (j [1, 9]), is the normalization constant of thefunction, denotes which players move is described by theGaussian, is the full width at half max of the Gaussianfunction and x0,,j is the center of the board square j. Deningeach board square to be of unit length, then j

x {0.5, 1.5, 2.5};jy {0.5, 1.5, 2.5}; = 0.2; and = 1/[ (2)1/2]. In this

scheme the center of the fth board square would be (5x, 5

y) =(1.5, 1.5). By using Gaussian functions to represent the wavefunction describing a players move, we aorded an opportunity

Figure 3. The measurement on a specic board can have observableramications even for game pieces that are not members of the cyclicentanglement; these pieces are referred to as dangling markers.



to teach the Gaussian integrals that are vital in quantumchemistry22 while exploiting the ease of the integral forms.23

Students seem to take to this introduction to the use ofGaussian functions more so than a typical introduction inatomic or molecular calculations. This may be due to the lessintimidating or esoteric application.One could assign to Alice a normalized wave function that is

a Gaussian-type function for her pieces with a negative ()leading sign and to Bob a Gaussian-type wave function withpositive (+) leading sign. Beginning a classical game of tic-tac-toe, allow both Alice and Bob to make their rst move. Bothplayers will recognize that the X and O represent game pieces,yet they have opposing signs. This will frame a discussion of thesign invariance of the wave function. During this discussion,these authors have found it appropriate to emphasize that it isthe magnitude of the functions displacement from zero that isof signicance and draw an analogue to waves in uids whilepointing out that the Laplacian term of the Schrodingerequation is used to describe uid waves as well.As these probability amplitudes can dier in both sign and

complexity (real vs imaginary), it is here that these authors haveintroduced the magnitude (in fact, the squared magnitude) tothe students as the valuable and physically interpretablequantity. As the function is possibly complex, one shouldremind the student that magnitude of a general complexnumber is given by |z| = (zz*)1/2 and that the wave functionacts in a similar fashion. We may now introduce the probabilitydensity, ||2, of the system as the physical quantity.In both the quantum and classical analogues of tic-tac-toe,

the system could either be described through a series of singleplayers moves, |i

j or the total state of the board,. In termsof the classical game, each move represents a complete particleon the board. These single particles each inhabit a principalsquare within the board, in this manner any function describinga specic particle would be linearly independent of a functiondescribing another. This example can be seen in Figure 4A; thislinearly independent set of moves can be described through thefollowing function for the total state of the board:

= | | | 1X

1 2O

5 3X

9 (2)

Similarly, a spooky marker represents a single particle thatexists in two dierent board square simultaneously and the

moves seen in Figure 4B can be described through a total boardwave function:

= | | |

= | + | | + | | + | 18( )( )( )

1X

2O

3X

1X

1 1X

2 2O

5 2O

6 3X

7 3X

9

(3)

We reserve explaining the factor of 1/8 to the student untillater.Our decision to use Gaussian functions lends itself to

instruction of these introductory concepts through CTTTalone; this allows the instructor to choose to reserve the use ofQTTT for times when it is more comprehendible to thestudent and more necessary for the course material. Theinstructor can choose to show that a classical game piece isrepresentable by a Gaussian function that can be of either sign.Both signs equally represent a particle and lead to a properlysigned (+) probability density for the system. At this point it isalso at the instructors discretion to employ imaginaryexponents in the Gaussian functions to show a properly signedmagnitude for the probability density and proving the need fortaking the complex conjugate of the wave function.

THE INNER PRODUCT, NORMALIZATION, ANDOVERLAP

Extending the discussions framed within the previous sectionallows for the introduction of the inner product whose generalform is:

| = * ( ) ( ) ( ) ( ) de (4)where refers to all coordinates within the function and e isthe bounds of the space dened by a specic problem. Theinner product may be exercised within the connes of the gamein ways that exemplify its two early uses: the normalization andthe overlap.Many early students beginning their studies in quantum

mechanics nd that the rst hurdle to their understanding isnormalization. We have used this game and presented methodsto successfully introduce this topic to students who arestruggling in their undergraduate quantum mechanics course;the authors feel that the student benets from the initialremoval of the concept from atomic and molecular systems.This allows the student to understand the concept intuitively,learn the mathematical statement and then transplant all of thisback into quantum mechanics. Starting with the boardsexpressed in Figure 5, we have used a series of activities totest the students comprehension of normalization.Students, from experience, recognize that when a classical

marker is placed in a square of the game board the marker iscompletely contained within that space and does not existwithin any other space on the board. In an eort to prove thatwhich the student already knows, we can perform the followinginner product using the wave function for just the X in Figure5A:

| = * g x y g x y x y( ) ( ) ( ( , )) ( , ) d dX X 5,5 5 5e (5)The inner product will be evaluated three times for Figure

5A. For the rst evaluation of eq 4 we shall dene e = Board;in this instance the students intuition that the marker issomewhere within the board is veried through the value of the

Figure 4. (A) Board shows a series of classical markers; by their natureof classical markers, any wave function describing one is linearlyindependent with any other markers wave function. (B) Board showsa series of spooky markers. The wave function describing this series ofmoves reveals that these partials are linearly independent with eachother.



integral being 1, thus permitting the student to solve for byfollowing intuition. We can further impress upon the studentthis point by the reevaluation of eq 4 with e = 5 and thenagain with e = 9 The rst of these evaluations again leads thestudent to accept that the marker is exactly where they think itshould be, in square 5. The later of these two activities merelyshows the student that the marker that is not in square 9.Shifting focus to evaluations of eq 3 on the board shown in

Figure 5A, we can now generate the linear combination, 1X =

1/2(|1X1 + |1X4) describing the state of spooky marker in amanner consistent with eq 3. Reverting to Dirac notation andthe students intuition, we can complete the followingsimplications and evaluations with e = Board:

| = | + | | + |

= | + | + | + |

= + + + =

14( )( )

12( )

12(1 0 0 1) 1

1X

1X

1X1 1

X4 1

X1 1

X4

1X

1X

1,1 1X

1X

1,4 1X

1X

4,1 1X

1X

4,4

(6)

The students by now have recognized that a spooky markerhas the same weight as a classical marker in the totality of theboard. These authors also chose to commit the inner product ofthe spooky marker in Figure 5 with e = 4, revealing thatsquare 4 contains half of the spooky marker.In a similar fashion, the instructor can impress both the

meaning and mechanism of the overlap integral onto thestudent through activities denable on game boards. Here, theuse of the Spooky Marker in this exercise is highlighted as theyare capable of overlapping with other spooky markers. Theprovided board and marker combinations in Figure 5 hold thepotential for a variety of activities for the student. Board 5Aworks as an example of the dierence between spooky markersand classical markers. Board 5B can be used to instruct overlapof both types of markers.

SEPARABILITY AND ENTANGLEMENTIf an instructor wishes to introduce the concept ofentanglement within the course, as we did, they may do soby introducing the most fundamental necessity for entangle-ment: inseparability of wave functions.2426 To this end, aseries of moves can be shown to the student, such as those seen

in Figure 6. As the game hinges on the generation of theentangled cycles through generation of inseparable states

through marker placement, this is a great opportunity toforge into this topic.It can be shown that the moves in Figure 6 are linearly

independent as the series of moves fails to generate a statewhose collapse into classicality is forbidden. This is claried byexample, observe Figure 6A; this series of moves can bedescribed by the following expression for the wave function ofthe board ():

=

= | + | | + |

= | | + | | + | | + | |

12( )

12( )

14( )

1X

2O

1X

1 1X

5 2O

6 2O

9

1X

1 2O

6 1X

1 2O

9 1X

5 2O

6 1X

5 2O

9

(7)

Here we pointed out to the students that the density of particleX is not cohabitating with any fraction of the density of particleO; this indicates that the classically collapsed state of particle Xhas no eect on the classically collapsed state of particle O. Theexpanded total state expression seen in equality 3 of eq 6 can berecollected back into equality 2; this state function can be saidto display the property of separability imbued on systemscomprised of states that are linearly independent of each other.This linear independence is forfeit if density fractions of thetwo particles share the same state (or position on the board), asseen in Figure 6B and whose functional description is here:

= | | + | | + | | 13( )

1X

2O

1X

1 2O

5 1X

1 2O

9 1X

5 2O

9(8)

It is clearly noted that the expanded form of the statesdescribing the board in eq 8 includes states that are forbiddenon the board, noted by the loss of the |1

X5 |2O5 state, which is

classically forbidden. Due to the loss of this mathematical state,the expression cannot be recollected as a product of the twoindividual moves; this is referred to as inseparability offunctions is a fundamental property for systems that possessand exhibit entanglement. Similarly, individual electrons can be

Figure 5. Boards that can be used during class activities: (A) a boardgiving a brief pair of activities that can be used to enforce the conceptof normalization as the student integrates the board over each of themarkers and then the pair of spooky markers and (B) a board yieldingseveral activities that can be used as a means of both enforcing theconcept of overlap and allow the student to numerically evaluation theoverlap integral of Gaussian-type functions.

Figure 6. (A) A pair of moves are placed in such a way, that the overallwave function of the board is separable; this can be shown through anexpansion of the product of the wave functions for each spooky markerand then the subsequent concretion back to the original product. (B)A pair of moves are placed whose total board wave function isinseparable; there exist members of the product expansion who areexclusionary to other members.



in the | state or the | state, yet when in a coupled pair, theelectron system can only be in the | state or the | state,noting the loss of the | state and the | state.

CONCLUSIONIn summary, we have presented a series of activities that may beused during introductory quantum mechanics and physicscourses. These activities have, through the experience of theseauthors, aided students in their understanding of quantummechanics by providing a degree of intuition to themathematics of the topic. This intuition provided by bothclassical and quantum versions of a childrens game with whichmost student have had some experience has beneted theinstruction simple topics within the course, especially normal-ization and simple statements described through the use ofwave functions. Furthermore, by exploiting the game we havefound this method lowers the degree of fear some studentspossess toward quantum mechanics. It is the hope of theseauthors that utilizing such intuitive examples may become aswidely accepted as has the use of the particle-in-a-box problem.These authors also hope that the armory of quantum gamesused in the classroom will be expanded to include otherversions of tic-tac-toe27 and furthered to a larger variety ofgames.6,28 A playable online version, which includes an AIplayer, for use by the students or practice for the instructor canbe found on the Web.29

ASSOCIATED CONTENT*S Supporting InformationBoard and example activities. This material is available via theInternet at http://pubs.acs.org.

AUTHOR INFORMATIONCorresponding Author

*E-mail: [email protected]

The authors declare no competing nancial interest.

ACKNOWLEDGMENTSThis work is supported by the NSF Centers for ChemicalInnovation: Quantum Information for Quantum Chemistry,CHE-1037992. The authors would also like to thank GeorgeBodner of Purdue University for critical reading.

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