In the Classroom edited by
Tested Demonstrations
Ed Vitz
The M&M® Superposition Principle submitted by:
John B. Miller Department of Chemistry, Western Michigan University, Kalamazoo, MI 49008;
[email protected] checked by:
Charles A. Smith Department of Chemistry, Our Lady of the Lake University, San Antonio, TX 78307
Kutztown University Kutztown, PA 19530
Brian W. Williams Department of Chemistry, Bucknell University, Lewisburg, PA 17837-2005
It is well known that undergraduate and graduate students find it difficult to grasp many fundamental concepts of quantum mechanics. This is particularly true when they are first exposed to the subject’s rigorous mathematical underpinnings. This first exposure generally occurs in a physical chemistry course during the junior or senior year. Introducing abstract wave-function descriptions of the ball-and-stick models to which they have become accustomed often comes as a rude awakening, just as the student was certain that chemistry was a manageable subject, consisting of reaction schemes, concentration determinations, and spectroscopic measurements. Many undergraduate students are understandably either unprepared or unwilling to tackle such abstractions, preferring concrete examples. As a discipline, chemistry relies heavily on demonstrations and models in instruction and research. However, since we inhabit a generally classical realm, appropriate physical demonstrations of quantum principles are limited, and computer simulations are most prevalent (1). This paper describes a physical system for demonstrating operators, eigenvalues, and superposition of states for a set of wave functions, albeit unusual ones. These topics seem especially baffling to many students, even though the mathematics involved is usually straightforward. In particular, students find that determining eigenfunctions of operators, discriminating between eigenvalues and expectation values, and linking operators to observables is an almost overwhelming task. Yet these topics are fundamental to understanding quantum mechanics well. Wave Functions, Operators, Eigenvalues, and All That In quantum mechanics, particles and particle systems are described as a single wave function ψ or a normalized linear combination of wave functions ψ = Σ c i ψi i
where the constants ci represent the relative contribution of each component ψi to the sum and
1 = Σ ci
2
i
Although wave functions are merely mathematical functions describing physical entities, we tend to treat the functions as
the physical items themselves. While useful, this approach puzzles many students until they have enough experience to draw the distinction themselves. To extract observable quantities from a wave function, ˆ, it is advantageous to use operators. Applying an operator Ω itself a mathematical function, generally consists of simple left multiplication of the wave function by the operator. If the resulting product is a constant multiple of the original ˆ = ωψ, ψ is said to be an eigenfunction of wave function, Ωψ ˆΩ with the constant ω its eigenvalue (characteristic value). Effectively, the operator “asks” the wave function to report a property about itself. A classic example wave function is a simple traveling wave restricted to one dimension (x) so that ψ(x) = e ikx, with ⁄ where i is √— its wavelength determined by k = p/h, ᎑1, p is the particle’s momentum, and h⁄ is Planck’s constant divided by 2π (2). To determine this wave function’s momentum, we ⁄ may apply the momentum operator pˆ = ( h/i)(d/dx), giving ˆp ψ = k h⁄ ψ, or a momentum eigenvalue ω = k h⁄ . Most students follow this introductory treatment reasonably well. Unfortunately, the principle of superposition strains many students’ understanding. This is a useful but abstruse description of particles as the superposition (or linear combination) of component wave functions, each potentially having a different observable value and each independently measurable by application of an operator. Although superposition can be graphically depicted fairly rigorously, as plots of interfering functions, for example, applying the abstraction of operators to an equally abstract function often does little but increase the glassiness of the students’ gaze. A particle confined to a one-dimensional box is a common example of superimposed quantum states (3). Such a particle can be described by the wave function ψn(x) ⬀ sin(nkx), where n (= 1, 2, 3, …) is the principal quantum number. Applying ⁄ the momentum operator yields pˆ ψn(x) = (nk h/i) cos(nkx), which is clearly not a constant multiple of the original wave function. Hence ψn is not an eigenfunction of pˆ. However, this standing wave is equivalent to the superposition of two traveling waves, ψ+ = einkx and ψ᎑ = e᎑inkx, each of which is individually an eigenfunction of pˆ, giving eigenvalues pro⁄ respectively. Quantum superposiportional to +k h⁄ and ᎑k h, tion says that individual measurements of the momentum of such a standing wave would give a random sequence of equal but opposite momenta, averaging to zero over a large number of observations.
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In the Classroom
Thus even wave functions that are not eigenfunctions of a given operator can provide observable properties for that operator provided the wave function can be described as a superposition of eigenfunctions. This aggregate observable 〈Ω〉, the expectation value of the wave function for the operator, is equal to a weighted average of the eigenvalues of the component superimposed eigenfunctions. The probability of selecting any particular component ψi out of the superposition of states is equal to |ci | 2. M&M’s as Wave Functions Giving wave functions a macroscopic physical manifestation is helpful, since it satisfies many students’ preference for a concrete model. The choice of the physical model is important. To give sufficient richness to a demonstration system, the wave function models should have a variety of properties—some in common and some different. It is also helpful if the models are particle-like and can be assembled within a volume not too dissimilar from the individual “wave functions.” The models should also be memorable. We have found M&M’s to serve very well as wave function surrogates.1 M&M’s are a popular family of small candies available in several varieties throughout North America. The basic design is a thin, hard shell, variously colored, imprinted with a lower case Latin m, and containing either milk chocolate (“Plain”), milk chocolate and a nut (“Peanut” or “Almond”), or peanut butter (“Peanut Butter”). The Plain and Peanut Butter varieties are small oblate spheroids, approximately 1 cm and 1.5 cm in diameter at the equator, respectively. The Plain candies are also available in a “Minia-
ture” size and are produced in two versions, containing either semisweet or milk chocolate. The nut-containing varieties are small prolate spheroids, approximately 1 cm at the equator and 1.5 cm along the major axis. The same company produces a smaller family of similar products called Skittles, which are colored, oblate spheroids imprinted with a lower-case Latin s, do not contain chocolate or nuts, and are available in various assortments of fruit flavors. Similar candies are available from other manufacturers and may of course be substituted (with appropriate modifications to specific operators, etc.), as could combinations of gum balls, sour balls, jawbreakers, or jelly beans, to name a few. An admixture of several varieties of dried beans, dried lentils, coins, or marbles could also be used, although perhaps less successfully, since their mnemonic value is somewhat lower. These objects meet many of the requirements for model wave functions: many related similar and dissimilar properties; small, particle-like nature; and memorability, perhaps the most important. Three features increase the memorability of this demonstration. First is the candies’ familiarity and the general fondness most students have for them. Second is the apparent absurdity of using candy in a quantum mechanics demonstration. Third is the candy’s edibility. Although consumption of food in a laboratory setting should be scrupulously avoided, eating the materials at the end of this demonstration, presumably performed in a lecture setting with uncontaminated materials and containers, provides a Pavlovian learning reward not to be underestimated, particularly if the lecture session is right before lunch. Given this effect, the mnemonic value of the inedible examples may be somewhat lower than that of the edible ones.
Table 1. Example Operators, Eigenvalues, and Expectation Values for M&M’s and Related Wave Functions ˆ Wavefunction ψ = Σψi Operator Ω Eigenvalue(s) ωi Expectation Value 具Ω典
880
Plain
Letter?
m
m
Peanut
Letter?
m
m
Plain + Peanut
Letter?
m, m
m
Skittles
Letter?
s
s
M&M’s + Skittles
Letter?
m, s
Latin letter
Plain
Shape?
oblate spheroid
oblate spheroid
Peanut
Shape?
prolate spheroid
prolate spheroid
Plain + Peanut
Shape?
prolate, oblate
"sphere"
Plain
Color?
brown, tan, green, red, blue, yellow, orange "black"
Peanut
Color?
brown, green, red, blue, yellow, orange
"black"
Skittles
Color?
red, orange, yellow, green, blue, violet
"black"
1:1 blue:yellow M&M’s Color?
blue, yellow
green
green M&M’s
Color?
green
green
1:2 blue:green M&M’s
Color?
blue, green
aqua
Plain
Shape?
oblate spheroid
oblate spheroid
Peanut
Shape?
prolate spheroid
prolate spheroid
Plain + Peanut
Shape?
prolate, oblate
"sphere"
Plain + Skittles
Shape?
oblate, oblate
oblate spheroid
Plain + Peanut Butter
Shape?
oblate, oblate
oblate spheroid
Plain
Equatorial diameter? ~1 cm
~1 cm
Peanut
Equatorial diameter? ~1 cm
~1 cm
Plain + Peanut
Equatorial diameter? ~1 cm, ~1 cm
~1 cm
Plain + Peanut
Polar Diameter?
~1 cm
~0.5 cm, ~1.5 cm
Journal of Chemical Education • Vol. 77 No. 7 July 2000 • JChemEd.chem.wisc.edu
In the Classroom
The Demonstration In front of the class, the demonstrator places selected assortments of M&M’s (or other items) into an opaque container while describing the process. This creates a superimposed system of “quantum states” ψ = Σn i ψ i, where ni is the number of each type of item in the container. The opacity of the container reduces later observer bias and aids in concealing that the system is a non-normalized ensemble. The demonˆ to strator or members of the class then suggest an operator Ω act on ψ . If all the ψ i provide the same eigenvalue ω, then ψ ˆ . However, if not may be proclaimed an eigenfunction of Ω all ωi are the same, then the individual ψ i should be polled by pulling out a single candy and determining its ωi, then returning the candy to the container and remixing. This should be repeated a statistically meaningful number of times and an estimate of the sample mean determined. A good minimum number of measurements would be at least 2(Σni)/ nmin (in analogy to detection of a signal frequency) where nmin is the number of the minority component of the mixture for the property being determined. Of course greater numbers of measurements will improve the statistics, but the students’ interest will likely wane quickly. Thus it would probably be best to keep the ratio of the minority component to the total greater than about one in three or four. The number of combinations of properties and operators is clearly very large, so an exhaustive list will not be attempted here. Nonetheless, several illustrative examples are shown in Table 1. The operators are posed as questions. Where the ψ are not eigenfunctions, an expectation value is estimated from the ωi. For some properties, such as color, averaging may not be as obvious, but very rapidly swirling a clear container of the candies can assist in visualizing the expectation value, as
can creating equivalent mixtures of pigments, such as poster paints. Conclusions Although the demonstration of superposition described above is not strictly accurate, it is quite effective and achieves a variety of educational goals. From the viewpoint of content, the students have a visual and concrete picture of a superposition of states, rather than an abstract plot of several overlaid mathematical functions. Operators are also firmly linked to observables and their underlying eigenfunctions, with a clear distinction made between eigenvalues and expectation values. Of greater importance is the pedagogical impact. The students have a good mnemonic for superposition: they will long remember having M&M’s in “P-chem”! Acknowledgment This paper is dedicated to the memory of the late Dr. Miles Pickering, who might have said, “Now, if we could only do this with beer, the Hobbits would really have it.” Note 1. M&M’s and Skittles are registered trademarks of the M&M/Mars Division of Mars, Inc., Hackettstown, NJ.
Literature Cited 1. For example: Pavlik, P. I. JCE: Software 1992, 5B, 23. Pavlik, P. I. JCE: Software 1992, 5B, 46. 2. Atkins, P. W. Physical Chemistry, 6th ed.; Freeman: New York, 1998; pp 299 ff. 3. Ibid.; pp 303 ff.
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