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American Journal of Physics

The hydrogen atom as a Morse oscillator

Soo‐Y. Lee

Published Aug 1, 1985• 10 citations
Publisher

Abstract

In its radial motion, the electron in the hydrogen atom can be pictured as oscillating about the Bohr orbits under Morse potentials. For each principal quantum number n, there corresponds a Morse oscillator with n bound states—one for each allowed value of the angular momentum quantum number l. The equilibrium positions of the Morse oscillators are exactly the Bohr orbits.

DOI
10.1119/1.14306

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In its radial motion, the electron in the hydrogen atom can be pictured as oscillating about the Bohr orbits under Morse potentials. For each principal quantum number n, there corresponds a Morse oscillator with n bound states—one for each allowed value of the angular momentum quantum number l. The equilibrium positions of the Morse oscillators are exactly the Bohr orbits.
The Bohr model is historically important in the understanding of quantum physics. It gives many important results for the hydrogen atom including atomic size and ionization energy. It is sometimes referred to as a semi-classical calculation, because it includes the physics of angular momentum quantization without the full quantum mechanical description of an electron wave. It is tremendously useful for gaining physical intuition of the quantum nature of the atom. In the Bohr model of the hydrogen atom, an electron has a circular orbit with speed v and radius r about the proton. The Bohr quantization condition is that an integer number of electron (de Broglie) wavelengths must fit to the atomic circumference.
Many teachers like to introduce the Bohr atom toward the end of an introductory physics course. This is an excellent idea, given the historic importance of Bohr’s 1913 work, which provided the bridge from Planck’s quantized interaction of matter and radiation (1900) to the full theory of quantum mechanics (1925-28). Unfortunately, the version of the Bohr atom that appears in many textbooks and is no doubt often presented to students is more wrong than right and may leave both teachers and students wondering why, more than a hundred years later, it is still being taught. This “pedagogic” version postulates that an electron in a stationary state moves in a circular orbit with an angular momentum that is an integral multiple of h/2π (L = nh/2π = nħ)—ħ for the lowest-energy state, 2ħ for the next state, and so on. This picture of the hydrogen atom is wrong in two senses. First it doesn’t conform to our present understanding of the hydrogen atom. This, in itself, is not a reason to scrap it, for the historical development of quantum physics is certainly of interest. But second, it doesn’t conform to the essence of what Bohr actually did. That is a reason not to teach about circular orbits and L = nħ.