Bohrgeschwindigkeit

Bohrgeschwindigkeit, in the German literature on atomic theory, refers to the speed of the electron in Bohr's early model of the atom. In this model the electron moves in circular orbits around the nucleus with a speed that depends on the orbital quantum number n and the nuclear charge Z.

The velocity in the Bohr model is commonly written as v_n = Z α c / n, where α is the fine-structure constant (approximately 1/137) and c is the speed of light. For hydrogen, where Z = 1, the ground-state velocity is v_1 ≈ α c ≈ 2.2 × 10^6 meters per second, about 0.7 percent of the speed of light. The speed decreases with increasing principal quantum number n and increases with higher nuclear charge Z, reflecting the stronger Coulomb attraction in more highly charged systems.

In Bohr's theory, the orbit radius scales as r_n = n^2 a0 / Z, with a0 ≈ 0.529 Å, and the angular momentum is quantized as L = nħ. The velocity relation thus implies that electrons in lower, more tightly bound orbits move faster than those in higher orbits.

Limitations and context: Bohrgeschwindigkeit is a characteristic of Bohr’s semiclassical model and is not a fundamental observable in modern quantum mechanics. In contemporary treatments, electrons are described by stationary states without a definite classical orbit or well-defined instantaneous velocity; the Bohr velocity remains a useful approximation for illustrating scaling in hydrogen-like systems and for historical understanding of Bohr’s model.