LETTERS TO THE EDITORS

# Quanta and Pennies

# Quanta and Pennies

The article "Quantum Identity" (May–June) by Peter Pesic contains two statements that may be misleading to readers.

The first statement is that Planck believed in "'atoms' of light, which could only mean discrete states of light energy, or quanta." Yet Max Jammer (*The Conceptual Development of Quantum Mechanics*, 1966) states that Planck viewed Einstein's 1905 "atomistic interpretation" of radiation—along with Einstein's rejection of the unrestricted application of Maxwell's electrodynamics—as "an as yet unnecessary step" and set out to resolve this situation "by a stricter adherence to Maxwell's electrodynamics." What Planck assumed was that the "material oscillators" were quantized while maintaining the classical view of the electromagnetic field. To this end Planck's "third theory" in 1914 regarded both emission and absorption by material oscillators as a continuous phenomenon with peaked transition probabilities.

The second statement is in the caption of Figure 4, which states, "Distinguishable quanta would have given a different answer, just as the number of possible outcomes in tossing two coins differs according to whether the coins are identical *(left) *or distinguishable *(middle)*." The figure shows two copper pennies—"identical objects"—in the left section and a copper and zinc penny—"distinguishable objects"—in the middle section. To test if the statistical outcomes of tosses for the two systems differed greatly, I used two copper pennies to mimic the "identical objects" case and a dime and a penny for the "distinguishable objects" case. The numerical distributions of the outcomes HH:HT:TT after 140 tosses of the coin pairs were 27:63:30 for the identical objects and 25:66:29 for the distinguishable objects. Both sets of pennies follow the 1:2:1 distribution, contrary to the caption in Figure 4.

The error in the caption is that the coins are macroscopic objects and the states of the pair of pennies can be defined by what the left or right columns reflect. Such left-right identification is applicable to both sets of pennies and therefore the distinguishable objects statistics apply to both. By limiting all the copper pennies to the left column and all the zinc pennies to the right column, the labeling of the "distinguishable" case is redundant.

Kenneth S. Schmitz

Department of Chemistry

University of Missouri, Kansas City

Planck's attitude towards his discovery was indeed complex and ambivalent, as I mentioned in my article. Dr. Schmitz rightly emphasizes Planck's attempt to forestall the radical implications of the quantum, even as late as 1914. However, already in 1909 Planck wrote that "we are forced ? to the conclusion that in heat radiation certain energy elements play an essential role." This is the basis of the account I gave in my article. Of course, Dr. Schmitz is also right that no ordinary coins would show the quantum statistics that Figure 4 tries to illustrate. For this reason, no figure can accurately render the peculiar behavior of radically indistinguishable "quantum coins." I regret if the figure or its caption misled any reader to think that any ordinary object can show the identicality of electrons and neutrons. In the spirit of analogy, the figure tries to indicate that two indistinguishable "quantum coins" have only three distinct outcomes, whereas two ordinary coins can have four.

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