Einstein's biggest mistake?

TL;DR

This paper examines Einstein's historical mistakes, suggesting his misnaming of A and B coefficients may have had significant long-term impacts on physics and chemistry education and understanding.

Contribution

It proposes that Einstein's misnaming of his A and B coefficients was a major overlooked mistake influencing multiple scientific fields.

Findings

Einstein's cosmological constant was later validated.

His rejection of quantum mechanics was influential but not a mistake.

The misnaming of A and B coefficients has affected scientific interpretation.

Abstract

What, if any, was Einstein's biggest mistake, the one most affecting our physics today? There is a perhaps apocryphal story, recounted by George Gamow, that he counted his cosmological constant as his biggest blunder. We now know his hypothesized cosmological constant to be correct. His lifelong rejection of quantum mechanics, an interesting side-story in the evolution of 20th-century physics, is a candidate. None of these introduced difficulties in how our physics is done today. It can be argued that his biggest actual mistake, one that affects many subfields of physics and chemistry and bewilders students today, occurred in his naming of his A and B coefficients.

Equations

$$dW=A_m^{n}dt$$

Full text

CORRESPONDENCE

To the Editors of ‘The Observatory’

What, if any, was Einstein’s biggest mistake, the one most affecting our physics today? There is a perhaps apocryphal story, recounted by George Gamow, that he counted his cosmological constant as his biggest blunder. O’Raifeartaigh and Mitton have recently1 summarized the available evidence and conclude that “We also find it quite plausible that Einstein made such a statement to Gamow in particular.” In any case, we now know his hypothesized cosmological constant to be correct2,3. Virginia Trimble has argued that his lifelong rejection of quantum mechanics, an interesting side-story in the evolution of 20th century physics, is a candidate. None of these introduced difficulties in how our physics is done today. It can be argued that his biggest actual mistake, one that affects many subfields in physics and chemistry and bewilders students today, occurred in his classic paper that first derived his $A$ and $B$ coefficients from considerations of the blackbody law and statistical equilibrium4. The Einstein $A$ is the rate of spontaneous emission of photons as an electron moves from an upper to lower energy level and is universally called his transition probability. Energy is conserved during the transition, so the photon energy is equal to the difference in energies between the two levels. Although electronic transitions produce most of the optical / UV / X-ray line emission, other transitions are possible. These include changes in the rotation or vibration of a molecular or the nuclear spin of an atom. The Einstein $A$ applies in all of these cases, so the idea has very broad applicability. In statistics, a probability is a dimensionless number between 0 and 1. Einstein’€™s $A$ has dimensions s*-1* and ranges between, for example, $\sim 3\times 10^{-15}$ s*-1* for the H I 21 cm line in the radio to $\sim 3\times 10^{14}$ s*-1* for the Fe XXVI K$\alpha$ line in the X-ray. Einstein’s $A$ is clearly not a probability. In his original 1917 paper, Einstein introduced his $A$ as follows:

Die Wahrscheinlichkeit dW dafut, dass dies im Zeitelement $dt$ wirklich stattfinde, sei

[TABLE]

wobei $A_{m}^{n}$ eine fur die betrachtete Indexkombination charakteristische Konstante bedeutet.

which can be translated as:

Let the probability $dW$ for this to happen during the time interval $dt$, be

[TABLE]

where $A_{m}^{n}$ is a constant characterizing the index combination under consideration.

The German word “Wahrscheinlichkeit” unambiguously translates to the English “probability”. The source of the confusion is the definition of $dW$. If $dW$ were really the probability over a time interval $dt$, then $A$ would be the probability per unit time, what we now call the “transition probability”. Actually, $dW$ is the number of photons emitted over the time $dt$. The $A$ is actually the rate photons are emitted, the Einstein transition rate. The term transition “probability” creates considerable confusion for students of astrophysical spectroscopy. Consider the quantities that enter into the description of how matter emits (section 3.5 of this5 text). Collisions can cause transitions within a two-level atom with upper and lower levels $u$ and $l$. At the most basic level, collisions between $u$ and $l$ are described in terms of a quantum-mechanical cross section $\sigma$ (cm2) which depends on the velocity $u$ (cm s*-1*) of the colliding particle. This cross section is integrated over a velocity distribution, usually Maxwellian, to obtain a rate coefficient $q_{ul}$ with the strange units cm3 s*-1*. The rate of collisional transitions $c_{ul}$ is $c_{ul}=q_{ul}n$ (s*-1*) where $n$ is the density of colliders (cm*-3*). The total rate that electrons move from $u$ to $l$ is then $r_{ul}=q_{ul}n+A_{ul}$. Knowledgeable students who understand dimensional analysis are bewildered by the concept of adding a rate and a probability. Although the term “transition probability” is an established part of the field6, the only part of quantitative spectroscopy which employs something like a probability involves many-level systems. The branching ratio from a level $u$ is defined as $R_{ul}=A_{ul}/\sum_{k<u}A_{uk}$. This is the probability that level $u$ will decay by the route $u\rightarrow l$, is dimensionless, and ranges between 0 and 1. $A$ is Einstein’€™s transition rate. Acknowledgements: Support by NSF (1816537), NASA (ATP 17-ATP17-0141), and STScI (HST-AR- 15018) is gratefully acknowledged.

Yours faithfully,

Physics & Astronomy,

The University of Kentucky,

Lexington,

Kentucky, USA, 40506

[gary@uky.edu]

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References

1. O’Raifeartaigh, C & Mitton, S. 2018, Physics in Perspective, Volume 20, Issue 4, pp.318-341

2. Perlmutter, S., et al., 1998, Nature, 391, 51

3. Riess, A.G. et al. 1998, AJ, 116, 1009

4. Einstein, A., 1917, Physikalische Zeitschrift, 18, 121-128

5. Osterbrock, D. E., & Ferland, G. J. 2006, Astrophysics of gaseous nebulae and active galactic nuclei, 2nd. ed. (Sausalito, CA: University Science Books)

6. The NIST online database of Einstein transition rates is accessible at

https://physics.nist.gov/PhysRefData/ASD/lines_form.html