A126252 Wavenumbers of red, turquoise, blue, indigo and violet in the spectrum of hydrogen, as first measured by Robert Bunsen and Gustav Kirchhoff in 1859.

1523310, 2056410, 2303240, 2437290, 2518130

Offset: 1

Author

Reinhard Zumkeller, Dec 22 2006

Keywords

• nonn
• fini
• full

Comments

How Johann Jakob Ballmer found his formula in 1885 by analyzing and manipulating the ratios of these data:

r(1) = a(1)/a(1) = 1,

a(2)/a(1) = 1.349961..., rounded: r(2) = 135/100 = 27/20,

a(3)/a(1) = 1.511996..., rounded: r(3) = 1512/1000 = 189/125,

a(4)/a(1) = 1.599996..., rounded: r(4) = 16/10 = 8/5,

a(5)/a(1) = 1.6530647..., r(5) = 81/49 = 2-1/(3-1/(9-1/2)), derived from a(5)/a(1) = 2-1/(3-1/(9-3095/6216)) when replacing 3095/6216 by 1/2;

the multiplication of these fractions by 5/36 is the key trick to get more handy figures to see eventually increasing squares in the denominators by an appropriate expansion:

b(1) = r(1)*5/36 = 5 / 36,

b(2) = r(2)*5/36 = 3 / 16,

b(3) = r(3)*5/36 = 21 / 100,

b(4) = r(4)*5/36 = 2 / 9,

b(5) = r(5)*5/36 = 45 / 196;

... b(1) .|.... b(2) ..|.... b(3) ..|.... b(4) ..|.... b(5),

... 5/36 .|.... 3/16 ..|... 21/100 .|.... 2/9 ...|... 45/196,

... 5/36 .|... 12/64 ..|... 21/100 .|... 32/144 .|... 45/196,

(9-4)/9*4 |(16-4)/16*4 |(25-4)/25*4 |(36-4)/36*4 |(49-4)/49*4,

this last step was the crowning achievement: the discovery of the pattern (x-y)/x*y,

b(n) = ((n+2)^2 - 4)/(4*(n+2)^2) = 1/4 - 1/(n+2)^2;

1<=n<=5: b(n) = A061037(n+2)/A061038(n+2) = A120072(n+2,2)/A120073(n+2,2).

References

• R. Taschner, Der Zahlen gigantischer Schatten, Vieweg 2005, 137-143.

Links

• Science Trek, Balmer Formula
• Eric Weisstein's World of Physics, Balmer Formula
• Wikipedia, Empirical formula
• Wikipedia, Johann Jakob Balmer
• Wikipedia, Robert Wilhelm Bunsen
• Wikipedia, Gustav Robert Kirchhoff