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A nearly 400 year-old formula for pi has been spotted in a quantum-mechanics formula for the energy states of the hydrogen atom, according to researchers in the US. Derived by English mathematician John Wallis in 1655, the original formula calculates pi as the product of an infinite series of ratios – it has now emerged from a solution of physicist Neils Bohr's early 20th-century hydrogen-atom model, which most budding physicists learn.

University of Rochester physicist Carl Hagen was designing homework problems for his graduate quantum-mechanics class when a particular exercise for the hydrogen atom intrigued him. It posed a twist on the Bohr model of hydrogen, which approximates the atom as an electron orbiting a point-like positive nucleus in circles. The Bohr model, while not an accurate description of an atom, is often close enough to the real thing in many situations. It is especially so when teaching physics, because it is one of the few systems that can be solved analytically by Schrödinger's equation – that is, it can be solved exactly, rather than making approximations or using a computer program.

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