D2.4 Hydrogen Atom: Quantum Mechanical Model

If we could calculate the energy for each energy level, we could predict the emission spectrum for hydrogen. In 1926, Erwin Schrödinger applied quantum mechanics, a model that uses both wave and particle analogies to describe atomic-scale matter, to the hydrogen atom. Instead of viewing the electron as a particle, Schrödinger applied mathematics appropriate for three-dimensional stationary waves constrained by electrostatic potential (Coulomb’s-law attraction between electron and nucleus). For each wave he derived a mathematical function to describe the wave, a wave function. The wave function is typically designated by the Greek letter ψ.

Schrödinger showed that these wave functions could be used to calculate allowed energies of a hydrogen atom. The calculated energies are given by this equation:

 E_n = - \dfrac{k}{n^2},\;\;\; n = 1,2,3, \ldots

where the proportionality constant k = 2.179 × 10−18 J, and n is a quantum number restricted to positive integer values.

In an electronic transition, an electron moves from an initial energy level, with energy Ei, to a final energy level, with energy Ef. The energy difference between the two energy levels is:

 \Delta E = E_\text{f} - E_\text{i} = - k\left(\dfrac{1}{n_\text{f}^2} - \dfrac{1}{n_\text{i}^2} \right)

A positive ΔE means that the atom’s energy increased, corresponding to absorption of a photon: the photon’s energy has been added to the atom’s initial energy. Similarly, a negative ΔE means that the atom has lost energy through emission of a photon.

Conservation of energy requires that the energy of the photon, Ephoton = hc/λ, equals the absolute value of the energy difference, |ΔE|, for emission or absorption. The sign of ΔE indicates whether the photon was absorbed (+) or emitted (−).

Exercise: Hydrogen-Atom Electronic Transition Energy

The equation Schrödinger obtained is equivalent to the equation Rydberg used to calculate hydrogen emission lines:

 \begin{array}{rcl} E_\text{photon} &=& |\Delta E| \\[0.5em] \dfrac{hc}{\lambda} &=& \left| -k \left( \dfrac{1}{n_\text{f}^2} - \dfrac{1}{n_\text{i}^2} \right) \right| \\[1.25em] \dfrac{1}{\lambda} &=& \dfrac{k}{hc} \left( \dfrac{1}{n_\text{f}^2} - \dfrac{1}{n_\text{i}^2} \right) = R_\infty \left( \dfrac{1}{n_2^2} - \dfrac{1}{n_1^2} \right) \end{array}

This relationship shows that R = k/(hc). Substituting values for k, h, and c into this equation gives R = 1.097× 107 m−1, which is the same as the experimental Rydberg constant to four significant figures. That a wave model could reproduce these highly accurate energy levels was strong evidence that quantum mechanics is an appropriate atomic-level model.

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Chemistry 109 Fall 2021 Copyright © by John Moore; Jia Zhou; and Etienne Garand is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.