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.

## Physical Concepts and Game Play

_{1}) exists in both squares 1 and square 5; the second player’s, referred to as Bob, first move (O

_{2}) exists both within square 5 and square 7; and Alice’s second move (X

_{3}) 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 X

_{1}shares principal square 1 with X

_{3}, X

_{3}shares principal square 7 with O

_{2}, and finally O

_{2}shares principal square 5 with X

_{1}. This series of moves is shown in Figure 2 and will be made clearer in a sample game.

_{i}

^{η}⟩

_{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, |ψ

_{1}

^{X}⟩, is a super position with the form: |ψ

_{1}

^{X}⟩ = 1/√2(|φ

_{1}

^{X}⟩

_{1}+ |φ

_{1}

^{X}⟩

_{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 ψ

_{1}

^{X}, ψ

_{2}

^{O}and ψ

_{3}

^{X}through their possible states (squares 1, 5, and 7).

_{3}

^{X}⟩ = 1/√2(|φ

_{3}

^{X}⟩

_{1}+ |φ

_{3}

^{X}⟩

_{7}), should be observed in square 7, this would imply that the only state that |ψ

_{2}

^{O}⟩ = 1/√2(|φ

_{2}

^{O}⟩

_{5}+ |φ

_{2}

^{O}⟩

_{7}) could take is that of square 5 and thus the only state |ψ

_{1}

^{X}⟩ = 1/√2(|φ

_{1}

^{X}⟩

_{1}+ |φ

_{1}

^{X}⟩

_{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.

_{3}

^{X}would collapse in square 1 forcing ψ

_{1}

^{X}in square 1 and finally ψ

_{2}

^{O}in square 5. The lower board is that which would occur if Bob chose to observe ψ

_{3}

^{X}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.

## Activity

## Probability Amplitude, Sign Symmetry, and Probability Density

*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.

*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

*x*

_{0,,j}is the center of the board square

*j*. Defining each board square to be of unit length, then μ

_{j}

^{x}∈ {0.5, 1.5, 2.5}; μ

_{j}

^{y}∈ {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 (μ

_{5}

^{x}, μ

_{5}

^{y}) = (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.

*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.

_{i}

^{η}⟩

_{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)

## The Inner Product, Normalization, and Overlap

_{e}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.

_{e}= Ω

_{Board}; 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 Ω

_{e}= Ω

_{5}and then again with Ω

_{e}= Ω

_{9}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.

_{1}

^{X}= 1/√2(|φ

_{1}

^{X}⟩

_{1}+ |φ

_{1}

^{X}⟩

_{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 Ω

_{e}= Ω

_{Board}:(6)

_{e}= Ω

_{4}, revealing that square 4 contains half of the spooky marker.

## Separability and Entanglement

_{1}

^{X}⟩

_{5}|φ

_{2}

^{O}⟩

_{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.