# 8: The Hydrogen Atom

- Page ID
- 4519

The hydrogen atom is of special interest because the hydrogen atom wavefunctions obtained by solving the hydrogen atom Schrödinger equation are a set of functions called atomic orbitals that can be used to describe more complex atoms and even molecules. This feature is particularly useful because, as we shall see in Chapters 9 and 10, the Schrödinger equation for more complex chemical systems cannot be solved analytically. By using the atomic orbitals obtained from the solution of the hydrogen atom Schrödinger equation, we can describe the structure and reactivity of molecules and the nature of chemical bonds. The spacings and intensities of the spectroscopic transitions between the electronic states of the hydrogen atom also are predicted quantitatively by the quantum treatment of this system.

- 8.1: The Schrödinger Equation
- The hydrogen atom, consisting of an electron and a proton, is a two-particle system, and the internal motion of two particles around their center of mass is equivalent to the motion of a single particle with a reduced mass.

- 8.2: The Wavefunctions
- The solutions to the hydrogen atom Schrödinger equation are functions that are products of a spherical harmonic function and a radial function.

- 8.3: Orbital Energy Levels, Selection Rules, and Spectroscopy
- The orbital energy eigenvalues obtained by solving the hydrogen atom Schrödinger equation is negative and approaches zero as the quantum number n approaches infinity. Because the hydrogen atom is used as a foundation for multi-electron systems, it is useful to remember the total energy (binding energy) of the ground state hydrogen atom.

- 8.4: Magnetic Properties and the Zeeman Effect
- Electrons in atoms also are moving charges with angular momentum so they too produce a magnetic dipole, which is why some materials are magnetic. A magnetic dipole interacts with an applied magnetic field, and the energy of this interaction is given by the scalar product of the magnetic dipole moment, and the magnetic field.

- 8.5: Discovering Electron Spin
- We then have charge moving in a circle, angular momentum, and a magnetic moment, which interacts with the magnetic field and gives us the Zeeman-like effect that we observed. To describe electron spin from a quantum mechanical perspective, we must have spin wavefunctions and spin operators. The properties of the spin states are deduced from experimental observations and by analogy with our treatment of the states arising from the orbital angular momentum of the electron.

- 8.6: Other One-Electron Systems
- The quantum mechanical treatment of the hydrogen atom can be extended easily to other one-electron systems such as \(He^+\), \(Li^{2+}\), etc. The Hamiltonian changes in two places. Most importantly, the potential energy term is changed to account for the charge of the nucleus, which is the atomic number of the atom or ion, \(Z\), times the fundamental unit of charge, \(e\).

- 8.7: Spin-Orbitals and Electron Configurations
- The wavefunctions obtained by solving the hydrogen atom Schrödinger equation are associated with orbital angular motion and are often called spatial wavefunctions, to differentiate them from the spin wavefunctions. The complete wavefunction for an electron in a hydrogen atom must contain both the spatial and spin components.

- 8.8: Coupling of Angular Momentum and Spectroscopic Term Symbols
- The observation of fine structure in atomic hydrogen emission revealed that an orbital energy level diagram does not completely describe the energy levels of atoms. This fine structure also provided key evidence at the time for the existence of electron spin, which was used not only to give a qualitative explanation for the multiplets but also to furnish highly accurate calculations of the multiplet splittings.

- 8.E: The Hydrogen Atom (Exercises)
- Exercises for the "Quantum States of Atoms and Molecules" TextMap by Zielinksi et al.

## Contributors

David M. Hanson, Erica Harvey, Robert Sweeney, Theresa Julia Zielinski ("Quantum States of Atoms and Molecules")