In the Classroom JChemEd.chem.wisc.edu • Vol. 78 No. 1 January 2001 • Journal of Chemical Education 57 A Quantum Mechanical Game of Craps: Teaching the Superposition Principle Using a Familiar Classical Analog to a Quantum Mechanical System Patrick E. Fleming Department of Chemistry and Biochemistry, Arizona State University, Tempe, AZ 85287; pfleming@asu.edu Many students struggle with concepts in quantum chemistry. The physical interpretation of the wave function causes particular grief. The Born interpretation that the square of the wave function gives a probability density is useful. It also provides the criterion necessary for normalizing a wave function. Unfortunately, though, it is confusing. Scientists tend to prefer models that infer a cause-and- effect relationship rather than a probabilistic description of observation. Nonetheless, quantum mechanical systems may have several quantum states that can be observed at any one time. The quantity that can be observed is given by the eigenvalue to some quantum mechanical operator acting on the wave function. Each observable value has some associated probability that can be extracted from the wave function for the system. The difficulty is that classical systems can exist in only one given state. When a coin is tossed, it is assumed to land as either “heads” or “tails”. The quantum mechanical coin, however, would be said to be both heads and tails with equal probability for both. This is the foundation of the famous “Schrödinger’s cat” analogy (1). Common sense based on classical Newtonian physics precludes the possibility of the coin’s being both heads and tails. It must be one or the other. This was the root of Albert Einstein’s objection to quantum mechanics. In a letter to Max Born in 1926 (2), Einstein wrote discussing his famous objection to quantum mechanics. Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice. The merits of this objection are still the subject of active research. None the less, the dice themselves are interesting. Dice have the useful property that the measurement defined as reading the number of spots on the top face produces a result that is naturally quantized. Thus dice can be very handy devices in illustrating quantum mechanical properties of wave functions and operators (3). Let’s define a set of wave functions and an operator pertaining to the roll of one die. We don’t know the exact form of the wave function or the operator, but we can define them in terms some of their more important properties. First, the wave functions must be orthonormal so that  ψ m | ψ n  = δ nm where δ nm is the Kronecker delta, which has the property δ nm =  1 n = m 0 n ≠ m The “roll” operator will operate on the wave function and return an eigenvalue that is the number of spots showing on the top face of the die after it has been observed. There are six possible values that can be measured when a die is rolled, since the die can land with 1, 2, 3, 4, 5, or 6 spots showing. These values are the eigenvalues obtained when the operator operates on the function ^ R | ψ n  = n| ψ n  so n can take any integral value from 1 through 6. Further, since n corresponds to the physically observable measurement that is made on the system, it must be real valued. (Imagine how difficult it would be to play dice if the dice sometimes came up with imaginary values !) The superposition principle suggests that as in any other quantum mechanical system, any linear combination of these basis functions will satisfy the requirements of our system. Thus the general function Ψ = c n | ψ n  Σ n =1 6 will also satisfy the requirements of a wave function describing the die. The values of the coefficients c n can be determined through normalization (  Ψ| Ψ  = 1). Ψ| Ψ =  c n Σ n =1 6  ψ n |  c m Σ m=1 6 | ψ m   = c n c m ψ n | ψ m Σ m=1 6 Σ n =1 6 Using the orthonormality of the wave functions, the substi- tution  ψ n | ψ m  = δ nm can be made: Ψ| Ψ = c n c m δ nm Σ m=1 6 Σ n =1 6 The Kronecker delta causes the argument of the double sum to be zero except when m = n. This picks out a single term for each possible value of n. Ψ| Ψ = c n 2 Σ n =1 6 Also, all the coefficients must be equal because the probability of observing each value upon rolling the die is the same as for any other value. Ψ| Ψ =6c n 2 =1 and c n = 1 6 for all n. Further, the probability of observing any one value