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The problem of the golden ratio has been known for almost three millennia. The task is to divide a segment into two parts, each of length x and 1, respectively. If x:1 = (1 + x):x then this ratio is called the golden ratio and is usually designated ϕ. Its value is ϕ = [(1 + √5) ⁄2]. ϕ is closely related to various problems: the roots of x5 – 1 = 0, the existence of 5-fold symmetry axes, Fibonacci numbers (1), Penrose quasicrystals (2), etc. What can the hydrogen atom and the golden ratio possibly have in common? Let us consider the radial distribution function for the first excited state of the H-atom: P20(r) = r2R202(r), where R20(r) is the radial wave function for the 2s state. Knowing (3) that it is easy to prove that the graph of P20(r) has two minima (at r1 = 0 and r2 = 2a0, where the probability of finding the electron is 0) and two maxima, at r3 = 2(1 – 1⁄ϕ)a0 and r4 = 2(1 + ϕ)a0. Alternatively, an observer placed at the point r = 2a0 where P20(r) is zero, sees maxima that are (–2/ϕ)a0 and (2ϕ)a0 away! Intriguing result, isn’t it? Literature Cited- Walser, H. The Golden Section, translated from German by Peter Hilton; The Mathematical Association of America: Washington, DC, 2001.
- Petruševski, V. M., Kalajdzievski, S. M., Najdoski, Z. M. Chem. Educator 2003, 8, 358–363.
- Atkins, P. W. Physical Chemistry, 5th edition; Oxford University Press: Oxford, 1994; pp 421–460.
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