Crackpot Spammer Ch. 02byTaunus©
Disclaimer: The following short story is fictional. No resemblance to any person, living or dead, is intended or should be inferred.
Faustus begins his lecture on the newly revealed knowledge in quantum physics.
The two transcendental irrational numbers most occurring in mathematics are pi and e. "pi" being approximately 3.1415926535... and e being approximately 2.718281828.... On these two numbers hang most of the mysteries of mathematics. The first equation to consider is
e^(i*t) = cos(t) + i*sin(t), where i = Sqrt(-1) = (-1)^(1/2)
"i" is the so-called "imaginary number, where i*i = -1.
And we blissfully set t in the above equation to pi to get
e^(i*pi) = -1 ...or... e^(i*pi) + 1 = 0
Here pi, e, and i each make their cameo appearance. The parsimony of symbols and numbers is remarkable. Then of course, come the Gaussian distribution: e^(-x^2/2), which is normalized to be f(x) = e^(-x^2/2) / Sqrt(2*pi), which is also remarkable. It manages to include the irrational number Sqrt(2); it lacks the imaginary number. But consider the hybrid by inserting x = i*pi into e^(-x^2/2). This yields a number
e^(pi^2/2) = 139.0456367... [Eqn_1]
Now this number does crop up here and there in abstract number theory, but until 30 May 2011 it did not have an application in applied math or physics. Now let A = Fine Structure Constant and B = Electron g Factor. Then we observe the remarkable coincidence, to seven significant figures:
A + 1/A + B = e^(pi^2/2) [Eqn_2]
This gives an equation relating the Fine Structure Constant and the Electron g Factor. If we assume that "B" is both accurately and precisely known, then "A" is found from a quadratic equation.
x^2 + (B-e^(pi^2/2))*x + 1 = 0, something soluble by high school math.
Your assignment is to look up the best values for the Fine Structure Constant and the Electron g Factor and plug them into the equation [Eqn_2]. Given the Electron g Factor, determine the Fine Structure Constant. Assume [Eqn_2] is correct. What would be an accurate value for A?
The value of the measure of a particular physical constant may be accurate, or precise, or neither, or both. Let's see what these terms really mean. We start with a measurement system. If the measurement is close to the actual (or true) value of the constant, then the measurement system is said to be accurate or to accurately measure the constant's value. The precision of a measurement system is a measure of the reproducibility of the measurement system.
So, let's look back in history. The number pi was originally conceived of as about three; it was then more precisely approximated as a variety of rational numbers including 22/7, a common approximation in High School mathematics. A rational number is always precise---it is a mathematical entity. But it may fail to accurately determine the value of pi. The number 22/7, for example, is a repeating decimal fraction: 3.142857 142857 142857.... We have that |pi – 22/7| is 0.00126, or thereabouts.
For centuries, millennia even, pi was considered as a dimensionless physical constant, the ratio of the circumference of a circle to its diameter. Various rational numbers were given as approximate values of pi. It wasn't until the Seventeenth Century that science and mathematics realized that pi was a transcendental irrational number and that its value could be determined by infinite series or infinite products. Now the value of pi is known both accurately and precisely. It is undeniably a mathematical and not a physical constant.
Measurements of dimensionless physical constants may be precise but not accurate. In statistics, a Type I error is a false positive, a Type II error is a false negative, and a Type III is the error of measuring the incorrect entity with the measurement system. Precision is important; however, accuracy is essential.
Let's try values of the Electron g Factor in [Eqn_2] and solve for A. We have a good approximation for A = Fine Structure Constant as well as 1/A.
B = 2.00 A = 0.007297227944674 1/A = 137.0383394327
B = 2.002 A = 0.007297334450971 1/A = 137.0363393262
B = 2.0023 A = 0.007297350427184 1/A = 137.0360393102
B = 2.00231 A = 0.007297350959725 1/A = 137.0360293097
B = 2.002319 A = 0.007297351439013 1/A = 137.0360203092
B = 2.00231930 A = 0.007297351454989 1/A = 137.0360200092
B = 2.002319304 A = 0.007297351455202 1/A = 137.0360200052
B = 2.0023193043622 A = 0.007297351455221 1/A = 137.0360200048
Recommended Values A = 0.0072973525376 1/A = 137.035999679
From B = 2.0023 on, we have A = 0.00729735 and 1/A = 137.0360.
These numbers are sufficient for most calculations. It wouldn't take much perturbation to nudge these constants into a perfect fit. The CODATA values move about from time to time as technology changes and new systematic errors are discovered. It will be History that eventually makes the final decision.
The driving constant behind this equation is
e^(pi^2/2) = 139.0456365056... [Eqn_3]
This constant has no name. It should be named after the Muse who inspired it: Sharon Stone, the actor. This constant has a particular mystery about it. It is related to e^(-pi^2/2), a value in the function g(x) = e^(-x^2/2), where x = i*pi, i being such that i*i=-1.
Your next assignment is to find where e^(pi^2/2) appears in abstract number theory and, from there, make an educated guess as to how it might relate the Fine Structure Constant to the Electron g Factor.
6 June 2011 Taunus Trumbo