Postage stamp poses a Fermi problem.

Abstract

Enrico Fermi was well-known for giving his students outrageous problems that could be tackled with insightful back-of-the-envelope estimates, but it seems that the stamp just issued by the U.S. Postal Service on 29 September ([1][1]) presents it own problem. The problem has to do with what Fermi wrote on the board, and you don't have to be a nuclear physicist to figure it out. The stamp reproduces a photo of Fermi taken in front of a chalkboard at the University of Chicago on 26 March 1948. In an online search, my friend Chris Bergevin found the picture at the American Institute of Physics Emilio Segre Visual Archives. The Segre Archives has designated the original photo “Fermi A16” ([2][2]). In the upper left-hand corner of the stamp is part of a formula neatly written on the board, the full expression being out of the frame (indicated in the figure by a red circle on the stamp). A little digging with the marvelous staff at the Segre Archives turned up another photograph, “Fermi A15” (bottom photo), taken on the same day, at the same photo shoot, probably within a minute or two of Fermi A16 (the postage-stamp photo). And, there it is—Fermi has written the definition of α, the fine-structure constant ([3][3]). Well…sort of. Fermi has completely screwed it up, by interchanging the role of ℏ and e : The expression should have read α = e 2/ℏ c. At first, I was reluctant to believe that Fermi, author of the 4-vertex model, maestro of the neutron, the atomic pile, and other great ideas, could have committed a blunder of this magnitude. I considered other explanations: (i) Fermi didn't write the equations on the board. Nope, it's his handwriting. I compared it with characters from his handwritten notes ([4][4]) (see the right-hand inset in the figure; black and white are reversed to make the comparison). (ii) His α is another quantity. Highly unlikely. If you work out the units—(mass × length)3/2—they make no sense ([5][5]). Or (iii) Fermi was a prankster. Perhaps, but what is the joke, and is it funny? While pondering this last alternative, I ran into a friend, a distinguished professor at the University of Chicago, and he pointed out the obvious: “Fermi was just having a bad day. Trotted out in front of the camera, his memory playing tricks on him, he simply mis-regurgitated α. End of story. It could happen to anyone.” I think my friend is right, but one doubt still nags at me: How could Fermi have remembered the correct sign on all the terms of the Schrodinger equation, but have forgotten that the fine-structure constant is basically the electromagnetic coupling? Did he seamlessly merge the fine-structure constant with ℏ2/2m, the coefficient of the Laplacian in the Schrodinger equation? You'd have to be a neuroscientist to figure that one out. 1. [↵][6]Enrico Fermi was born on 29 September 1901. 2. [↵][7] . 3. [↵][8]Max Planck and Albert Einstein first noted that Planck's constant h had the same dimensions as e 2/ c and roughly the same order of magnitude. The constant α, later named and used concretely by Arnold Sommerfeld in atomic theory, is the dimensionless quantity that links the discreteness of electric charge ( e ), quantum theory ( h ), and relativity ( c ). 4. [↵][9]1. E. Fermi , Notes on Quantum Mechanics (Univ. of Chicago Press, Chicago, IL, 1995). 5. [↵][10]Multiply Fermi's expression with the square root of the true α; the dimensions do not change, but one gets (ℏ/ c )3/2. [1]: #ref-1 [2]: #ref-2 [3]: #ref-3 [4]: #ref-4 [5]: #ref-5 [6]: #xref-ref-1-1 View reference 1 in text [7]: #xref-ref-2-1 View reference 2 in text [8]: #xref-ref-3-1 View reference 3 in text [9]: #xref-ref-4-1 View reference 4 in text [10]: #xref-ref-5-1 View reference 5 in text