Chapter 12: Hyperfine Splitting in Hydrogen

To model this system, the discussion begins by establishing a basis set of four states which represent the combination of the electron having an "up" or "down" spin and the proton having an "up" or "down" spin, treating both as spin one-half particles. The core theoretical task involves defining the Hamiltonian for the hydrogen atom's ground state, which includes terms describing this crucial magnetic interaction using the sigma spin operators associated with the electron (sigma e) and the proton (sigma p). Solving the stationary state problem without an external field reveals four distinct energy levels. Three of these states are found to have a relatively higher energy (E equals A), while the fourth state, known as the singlet state, possesses a lower energy (E equals minus 3A). This calculated energy difference, Delta E, is highly significant, corresponding precisely to the 21-centimeter microwave line emitted by atomic hydrogen, which is a key signal utilized in the field of radio astronomy. The analysis is then expanded to include the effects of an external magnetic field (B field) on the energy levels, leading into the phenomenon of Zeeman splitting. The updated Hamiltonian incorporates the interaction of both the electron's and proton's magnetic moments with the external field, resulting in energy eigenvalues that depend on the magnetic field magnitude. The chapter illustrates how the stationary states evolve: while they are complex mixtures at zero magnetic field, they smoothly transition toward pure states aligned parallel or anti-parallel to the external field direction when the field strength is very large. Finally, the text explores the connection between the base states and total angular momentum quantum numbers (J and m), concluding with the formalism of the projection matrix for spin one, which helps translate the spin one-half particle system into a combined spin one system.