The learning of quantum mechanics is contingent upon an understanding of the physical significance of the mathematics that one must perform. Concepts such as normalization, superposition, interference, probability amplitude, and entanglement can prove challenging for the beginning student. Several class activities that use a nonclassical version of tic-tac-toe are described to introduce several 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 of superposition 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 other topics with the aid of proper discussion.
Classical mechanics, whose approach was developed based on Newton’s new mathematics, was contemporaneously formulated alongside calculus. Both topics moved from academic investigation into undergraduate lecture halls, and in the case of Newtonian mechanics, earlier still, with its concepts being introduced prior to high school. Quantum mechanics, developed in the 20th century, was required to adequately describe such experimental phenomena as blackbody radiation, the photoelectric effect, and the atomic spectrum of hydrogen. The development of quantum mechanics has led to description of phenomena such as the superposition principle, the ability of an unobserved quantum object to exist in a superposition of multiple states simultaneously; entanglement, spooky action at a distance where the state of one system affects that of another without a direct observable relationship connecting them; and interference, as matter exists in both particle and waveform within quantum theory, matter interactions present wave phenomenon such as diffraction and the properties of constructive and destructive matter–wave addition. Just as a rudimentary understanding, at minimum, of classical mechanics became necessary for so many fields, an introduction into the concepts of quantum mechanics is of growing importance.
A student’s first excursion into quantum mechanics can be both overwhelming and daunting, even to an upper-division science student. Understanding such concepts as wave functions, overlap integrals, and probability amplitudes are vital in mastering the subsequent material within the course. A typical first semester course in quantum mechanics focuses on the Schrödinger picture and equation.(1-3) Herein we present several activities using quantum tic-tac-toe (QTTT), which is a quantum analogue of classical tic-tac-toe (CTTT), presented by Allen Goff,(4-6) as a means of introducing and enforcing early topics in an introductory quantum mechanics course. The activities allow for introduction and discussion of probability amplitude, probability density, normalization, overlap, the inner product, and separability of states. It is the belief of the authors that QTTT can be used as an approachable, fun, and intuitive means of introducing these topics. It is the hope of the authors that this tool could act as a companion throughout instruction; after the students have been taught the game, the instructor can use it as a stepping-stone to new topics and as an avenue for intuitive activities.
The activities described, as well as other similar activities, have been used to assist the understanding of various audiences in anything from a brief understanding of concepts necessary to quantum computing to furthering a student’s understanding of topics in their quantum mechanics classroom. The bulk of the material was used as assignments and Supporting Information in an undergraduate quantum mechanics classroom to great avail with students who did not grasp some early concepts within the course.
Physical Concepts and Game Play
Board
The tic-tac-toe board is square and is divided into nine square subspaces. These subspaces will be referred to as principal squares and will each carry a number to denote the particular square being referenced. The numbering pattern of the principal squares on the board is shown in Figure 1. Prior to discussing the game play, some vocabulary and concepts are introduced. The following four elements are underlying physical concepts that are necessary for game play and thus their use is weaved within the description of the game.
Figure 1. The layout of the game board for either classical or quantum tic-tac-toe. This figure also displays the enumeration scheme that is used throughout this paper.
Spooky Marker
Named after Einstein’s reference to entanglement and hidden variable interactions as “spooky action at a distance”.(7) This is a direct consequence to the system being completely described through a finite number of basis functions of an observable. A coupled pair of electrons exist within a 0-spin state; that is to say that the wave function of the pair is of the form: ψ = 1/√2(|↑↓⟩ + |↑↓⟩). If one observes the state of a single electron within the pair, say it is in the up-state, that observer incidentally knows the state of the other spin within the pair. Like CTTT markers, the spooky marker represents a single move of a single player during one turn, yet a spooky marker exists within two separate principal squares simultaneously.
Superposition
As players have placed a pair of spooky markers that represents their move for that turn, this move can be said to exist as a superposition of the states (board positions) in which it may be realized. If player one, referred to as Alice, places spooky markers for her first move into squares 1 and 5, then the state of that move is the superposition of the two states: square 1 and square 5. All player moves within QTTT are superposition moves. A typical teaching example of this is the superposition of spins separated through observation in the Stern–Gerlach(8-10) experiments, which are typically discussed in introductory quantum mechanics courses. A further example, to which the students may have already been exposed, is the superposition of ammonia states by tunneling; the students may have discussed this already in their organic chemistry course with reference to nitrogen inversions(11) and the topic can be expounded through a discussion of the MASER problem.(12)
Cyclic Entanglement
Entanglement is the correlation between parts of a system, induced through an interaction and maintained in separation, which is independent of factors such as position and momentum.(13) In QTTT this would consist of a group of spooky markers whose board positions are all self-referencing; as an example, Alice’s first move (X1) exists in both squares 1 and square 5; the second player’s, referred to as Bob, first move (O2) exists both within square 5 and square 7; and Alice’s second move (X3) is within both square 7 and square 1. In this way, the possible states of these moves are dependent upon each other in a similar fashion as to the spin states of paired electrons. The cyclic reference here is that X1 shares principal square 1 with X3, X3 shares principal square 7 with O2, and finally O2 shares principal square 5 with X1. This series of moves is shown in Figure 2 and will be made clearer in a sample game.
Figure 2. The effect of “measurement” on the system of cyclic entanglement can yield, at minimum, a pair of classical states corresponding to the state in which X3 was observed.
State Collapse
A quantum system may exist in a superposition of several states. Only one subordinate state is observed when the state of the system is measured. An example of this would be a doublet spin system; the state of the single electron would be a superposition of up-spin and down-spin, yet when observed, a single electron will present only either an up-spin or a down-spin state. When a state collapse occurs through observation within the game, spooky markers collapse into 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 easily fleshed-out through an example game. Game play begins as Alice places her first pair of spooky markers on the board; any such move within the game will be denoted by |ψiη⟩j, where η represents the player to whom the marker belongs and will thus take on the values X or O, i denotes the turn when this marker was placed, and j is the location on the board where the marker was placed. She places her markers in principal squares 1 and 5. This means that her first move, |ψ1X⟩, is a super position with the form: |ψ1X⟩ = 1/√2(|φ1X⟩1 + |φ1X⟩5). Let us now have Bob place his markers in principal squares 5 and 7; unlike the classical tic-tac-toe game, the placement of a spooky marker in QTTT does not prevent either player from placing subsequent markers in a particular square. Alice retorts with markers in principal squares 7 and 1. With this last move, our game board is now consistent with that in Figure 2. It can now be seen that the state of each of the spooky markers is a linear combination of the two squares that it occupies and each position within this linear combination is a position within a linear combination describing another spooky marker. In Figure 2 it can now be seen that we have generated a cyclic entanglement between markers placed for ψ1X, ψ2O and ψ3X 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 a state collapse of our spooky markers into classical markers. As Alice’s last move was that which sealed the cyclic entanglement, it will be Bob’s right to decide in which way the states will collapse; this reciprocation of closure and observation was developed in hope to generate a fair game, although it was an ad hoc rule implemented for the sake of fair game play (a more quantum mechanically accurate rule would be flipping a coin to decide the collapse). When an observation is made on the system, the states of the markers involved with the cyclic entanglement will collapse. Unlike a spooky marker, when a classical 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 first state completely the logic of the upper path and then that of the lower path. If Bob chooses that Alice’s most recent move, |ψ3X⟩ = 1/√2(|φ3X⟩1 + |φ3X⟩7), should be observed in square 7, this would imply that the only state that |ψ2O⟩ = 1/√2(|φ2O⟩5 + |φ2O⟩7) could take is that of square 5 and thus the only state |ψ1X⟩ = 1/√2(|φ1X⟩1 + |φ1X⟩5) can manifest is that of square 1; all this due to the fact that this observation turns these spooky markers into classical markers and thus exclusively occupy their observed site.
If Bob had chosen the other path, ψ3X would collapse in square 1 forcing ψ1X in square 1 and finally ψ2O in square 5. The lower board is that which would occur if Bob chose to observe ψ3X 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 its states being enveloped by the cycle. In these cases, the observation will also effect a collapse upon the “dangling” marker; the subsequent collapse of dangling markers can also be seen in Figure 3.
Figure 3. The “measurement” on a specific board can have observable ramifications even for game pieces that are not members of the cyclic entanglement; these pieces are referred to as dangling markers.
Game play will continue in this manner until one of the players has generated a “three-in-a-row” consisting of only classical markers. It is possible that two players will simultaneously win the game through the same observation. When this occurs, the player with the most recent Spooky Makers generating one of their winning classical markers loses; which is, in a way, to say “first in, first out”. When playing a volley of games, Goff(4-6) does propose that the winning player during a simultaneous victory be awarded 1 point and the loser 1/2 point.
Activity
We present the following work as an instructor-guided inquiry activity(14, 15) with the students divided into groups of two. We present specific board examples as a means of discussion and instructional guidance examples, as introductory courses have been shown to benefit from strong instructor guidance.(16) A more natural experience would be allowing the students (postinstruction on the rules and teaching a specific phenomenon) to play the game and come across these phenomena on their own in an inductive learning style similar to a lab exercise.(17-20) QTTT could also be used as a continuing-themed homework exercise as it can be used to exemplify many of the introductory topics in quantum mechanics.
Introducing the game rules and running a small example game can take up to 15 min, whereas the average time to play a single game is roughly 4 min. In the experience of these authors, the use of quantum tic-tac-toe lowers the level of fear associated with introducing these early concepts, as it both builds student confidence and gives them a foothold on the material through a familiar mechanism. Students took to the game enthusiastically and divorced of the quantum mechanical concepts, learning the game rules comes quickly. The most difficult part in learning the game is recognizing the closed loops; it is suggested that the instructor select a student to act as a representative for all the students as the class plays against the instructor for a game; this method seems to reveal the present thought processes of the students, which can benefit instruction. These authors also found that the notions to be discussed within the following sections of this paper benefitted from introduction through QTTT as they are, at times, early signs 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, as discussion of pairs entangled particles uses the notation particles A and B; from this notation, observers at each end of the system are often referred to as Alice (for A) and Bob (for B).(13)
Probability Amplitude, Sign Symmetry, and Probability Density
The fundamental quantity within the Schrödinger picture of quantum mechanics is the wave function, Ψ(x). Ψ(x) are the solutions to the second-order differential wave equation describing the total system energy of a particle.(21) The use of either QTTT or CTTT does not lend itself to the introduction of the Schrödinger equation as there are no intuitive nor appropriate methods for the student to connect game play to energy. Yet use of QTTT has proven beneficial in the explanation and discussion of several properties of the wave function, especially topics such as normalization and sign symmetry of the probability amplitude.
Wave functions, as stated by the first 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 moves lends itself immediately as a means of emphasizing the sign invariance of the probability density. We will begin by defining:
(1)where j denotes the board space in which the Gaussian function resides (j ∈ [1, 9]), α is the normalization constant of the function, η denotes which player’s move is described by the Gaussian, μ is the full width at half max of the Gaussian function and x0,,j is the center of the board square j. Defining each board square to be of unit length, then μjx ∈ {0.5, 1.5, 2.5}; μjy ∈ {0.5, 1.5, 2.5}; σ = 0.2; and α = 1/[σ (2π)1/2]. In this scheme the center of the fifth board square would be (μ5x, μ5y) = (1.5, 1.5). By using Gaussian functions to represent the wave function describing a player’s move, we afforded an opportunity to teach the Gaussian integrals that are vital in quantum chemistry(22) while exploiting the ease of the integral forms.(23) Students seem to take to this introduction to the use of Gaussian functions more so than a typical introduction in atomic or molecular calculations. This may be due to the less intimidating or esoteric application.
(1)where j denotes the board space in which the Gaussian function resides (j ∈ [1, 9]), α is the normalization constant of the function, η denotes which player’s move is described by the Gaussian, μ is the full width at half max of the Gaussian function and x0,,j is the center of the board square j. Defining each board square to be of unit length, then μjx ∈ {0.5, 1.5, 2.5}; μjy ∈ {0.5, 1.5, 2.5}; σ = 0.2; and α = 1/[σ (2π)1/2]. In this scheme the center of the fifth board square would be (μ5x, μ5y) = (1.5, 1.5). By using Gaussian functions to represent the wave function describing a player’s move, we afforded an opportunity to teach the Gaussian integrals that are vital in quantum chemistry(22) while exploiting the ease of the integral forms.(23) Students seem to take to this introduction to the use of Gaussian functions more so than a typical introduction in atomic or molecular calculations. This may be due to the less intimidating 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 with positive (+) leading sign. Beginning a classical game of tic-tac-toe, allow both Alice and Bob to make their first move. Both players will recognize that the X and O represent game pieces, yet they have opposing signs. This will frame a discussion of the sign invariance of the wave function. During this discussion, these authors have found it appropriate to emphasize that it is the magnitude of the function’s displacement from zero that is of significance and draw an analogue to waves in fluids while pointing out that the Laplacian term of the Schrödinger equation is used to describe fluid waves as well.
As these probability amplitudes can differ in both sign and complexity (real vs imaginary), it is here that these authors have introduced the magnitude (in fact, the squared magnitude) to the students as the valuable and physically interpretable quantity. As the function is possibly complex, one should remind the student that magnitude of a general complex number is given by |z| = (z·z*)1/2 and that the wave function acts in a similar fashion. We may now introduce the probability density, |Ψ|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 single player’s moves, |ψiη⟩j or the total state of the board,Ψ. In terms of the classical game, each move represents a complete particle on the board. These single particles each inhabit a principal square within the board, in this manner any function describing a specific particle would be linearly independent of a function describing another. This example can be seen in Figure 4A; this linearly independent set of moves can be described through the following function for the total state of the board:
(2)
(2)
Figure 4. (A) Board shows a series of classical markers; by their nature of classical markers, any wave function describing one is linearly independent with any other marker’s wave function. (B) Board shows a series of spooky markers. The wave function describing this series of moves reveals that these partials are linearly independent with each other.
Similarly, a spooky marker represents a single particle that exists in two different board square simultaneously and the moves seen in Figure 4B can be described through a total board wave function:
(3)
(3)We reserve explaining the factor of 1/√8 to the student until later.
Our decision to use Gaussian functions lends itself to instruction of these introductory concepts through CTTT alone; this allows the instructor to choose to reserve the use of QTTT for times when it is more comprehendible to the student and more necessary for the course material. The instructor can choose to show that a classical game piece is representable by a Gaussian function that can be of either sign. Both signs equally represent a particle and lead to a properly signed (+) probability density for the system. At this point it is also at the instructor’s discretion to employ imaginary exponents in the Gaussian functions to show a properly signed magnitude for the probability density and proving the need for taking the complex conjugate of the wave function.
The Inner Product, Normalization, and Overlap
Extending the discussions framed within the previous section allows for the introduction of the inner product whose general form is:
(4)where τ̃ refers to all coordinates within the function and Ωe is the bounds of the space defined by a specific problem. The inner product may be exercised within the confines of the game in ways that exemplify its two early uses: the normalization and the overlap.
(4)where τ̃ refers to all coordinates within the function and Ωe is the bounds of the space defined by a specific problem. The inner product may be exercised within the confines of the game in ways that exemplify its two early uses: the normalization and the overlap.Many early students beginning their studies in quantum mechanics find that the first hurdle to their understanding is normalization. We have used this game and presented methods to successfully introduce this topic to students who are struggling in their undergraduate quantum mechanics course; the authors feel that the student benefits from the initial removal 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 this back into quantum mechanics. Starting with the boards expressed in Figure 5, we have used a series of activities to test the student’s comprehension of normalization.
Figure 5. Boards that can be used during class activities: (A) a board giving a brief pair of activities that can be used to enforce the concept of normalization as the student integrates the board over each of the markers and then the pair of spooky markers and (B) a board yielding several activities that can be used as a means of both enforcing the concept of overlap and allow the student to numerically evaluation the overlap integral of Gaussian-type functions.
Students, from experience, recognize that when a classical marker is placed in a square of the game board the marker is completely contained within that space and does not exist within any other space on the board. In an effort to prove that which the student already knows, we can perform the following inner product using the wave function for just the X in Figure 5A:
(5)
(5)The inner product will be evaluated three times for Figure 5A. For the first evaluation of eq 4 we shall define Ωe = ΩBoard; in this instance the student’s intuition that the marker is somewhere within the board is verified through the value of the integral being 1, thus permitting the student to solve for α by following intuition. We can further impress upon the student this point by the reevaluation of eq 4 with Ωe = Ω5 and then again with Ωe = Ω9 The first of these evaluations again leads the student to accept that the marker is exactly where they think it should be, in square 5. The later of these two activities merely shows 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, ψ1X = 1/√2(|φ1X⟩1 + |φ1X⟩4) describing the state of spooky marker in a manner consistent with eq 3. Reverting to Dirac notation and the student’s intuition, we can complete the following simplifications and evaluations with Ωe = ΩBoard:
(6)
(6)The students by now have recognized that a spooky marker has the same weight as a classical marker in the totality of the board. These authors also chose to commit the inner product of the spooky marker in Figure 5 with Ωe = Ω4, revealing that square 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 the student through activities definable on game boards. Here, the use of the Spooky Marker in this exercise is highlighted as they are capable of overlapping with other spooky markers. The provided board and marker combinations in Figure 5 hold the potential for a variety of activities for the student. Board 5A works as an example of the difference between spooky markers and classical markers. Board 5B can be used to instruct overlap of both types of markers.
Separability and Entanglement
If an instructor wishes to introduce the concept of entanglement within the course, as we did, they may do so by introducing the most fundamental necessity for entanglement: inseparability of wave functions.(24-26) To this end, a series of moves can be shown to the student, such as those seen in Figure 6. As the game hinges on the generation of the entangled cycles through generation of inseparable states through marker placement, this is a great opportunity to forge into this topic.
Figure 6. (A) A pair of moves are placed in such a way, that the overall wave function of the board is separable; this can be shown through an expansion of the product of the wave functions for each spooky marker and then the subsequent concretion back to the original product. (B) A pair of moves are placed whose total board wave function is inseparable; there exist members of the product expansion who are exclusionary to other members.
It can be shown that the moves in Figure 6 are linearly independent as the series of moves fails to generate a state whose collapse into classicality is forbidden. This is clarified by example, observe Figure 6A; this series of moves can be described by the following expression for the wave function of the board (Ψ):
(7)Here we pointed out to the students that the density of particle X is not cohabitating with any fraction of the density of particle O; this indicates that the classically collapsed state of particle X has no effect on the classically collapsed state of particle O. The expanded total state expression seen in equality 3 of eq 6 can be recollected back into equality 2; this state function can be said to display the property of separability imbued on systems comprised of states that are linearly independent of each other. This linear independence is forfeit if density fractions of the two particles share the same state (or position on the board), as seen in Figure 6B and whose functional description is here:
(8)It is clearly noted that the expanded form of the states describing the board in eq 8 includes states that are forbidden on the board, noted by the loss of the |φ1X⟩5 |φ2O⟩5 state, which is classically forbidden. Due to the loss of this mathematical state, the expression cannot be recollected as a product of the two individual moves; this is referred to as inseparability of functions is a fundamental property for systems that possess and exhibit entanglement. Similarly, individual electrons can be in the |↑⟩ state or the |↓⟩ state, yet when in a coupled pair, the electron system can only be in the |↑↓⟩ state or the |↓↑⟩ state, noting the loss of the |↑↑⟩ state and the |↓↓⟩ state.
(7)Here we pointed out to the students that the density of particle X is not cohabitating with any fraction of the density of particle O; this indicates that the classically collapsed state of particle X has no effect on the classically collapsed state of particle O. The expanded total state expression seen in equality 3 of eq 6 can be recollected back into equality 2; this state function can be said to display the property of separability imbued on systems comprised of states that are linearly independent of each other. This linear independence is forfeit if density fractions of the two particles share the same state (or position on the board), as seen in Figure 6B and whose functional description is here:(8)It is clearly noted that the expanded form of the states describing the board in eq 8 includes states that are forbidden on the board, noted by the loss of the |φ1X⟩5 |φ2O⟩5 state, which is classically forbidden. Due to the loss of this mathematical state, the expression cannot be recollected as a product of the two individual moves; this is referred to as inseparability of functions is a fundamental property for systems that possess and exhibit entanglement. Similarly, individual electrons can be in the |↑⟩ state or the |↓⟩ state, yet when in a coupled pair, the electron system can only be in the |↑↓⟩ state or the |↓↑⟩ state, noting the loss of the |↑↑⟩ state and the |↓↓⟩ state.Conclusion
In summary, we have presented a series of activities that may be used during introductory quantum mechanics and physics courses. These activities have, through the experience of these authors, aided students in their understanding of quantum mechanics by providing a degree of intuition to the mathematics of the topic. This intuition provided by both classical and quantum versions of a children’s game with which most student have had some experience has benefited the instruction simple topics within the course, especially normalization and simple statements described through the use of wave functions. Furthermore, by exploiting the game we have found this method lowers the degree of fear some students possess toward quantum mechanics. It is the hope of these authors that utilizing such intuitive examples may become as widely accepted as has the use of the particle-in-a-box problem. These authors also hope that the armory of quantum games used in the classroom will be expanded to include other versions of tic-tac-toe(27) and furthered to a larger variety of games.(6, 28) A playable online version, which includes an AI player, for use by the students or practice for the instructor can be found on the Web.(29)