Summary
Highlights
Hyperfine splitting is analogous to spin-orbit coupling, but it arises from the interaction between the spin of the proton and the spin of the electron. This effect is significantly smaller than the fine structure corrections, hence the name 'hyperfine.' To understand this, we consider the proton as a magnetic dipole, which generates a magnetic field. The interaction of the electron in this magnetic field contributes to the Hamiltonian.
The hyperfine Hamiltonian is given by the negative dot product of the electron's magnetic dipole moment and the magnetic field produced by the proton. The electron's magnetic dipole moment is known from previous derivations related to fine structure. The magnetic field produced by a magnetic dipole moment is quoted from electromagnetic theory. The proton's magnetic dipole moment is taken from literature, as it's a composite structure, unlike the electron.
The derived Hamiltonian is complex. To find the energy levels, we need to apply degenerate perturbation theory, which can be simplified if we find the 'good states' of the system. These states are based on the total spin, which is conserved. We then take the expectation value of the hyperfine Hamiltonian, which will give us the energy. This involves evaluating several expectation values, including terms involving the radial distance and dot products of spin vectors.
The problem asks to prove that an integral involving dot products of constant vectors with the radial unit vector, integrated over spherical coordinates, evaluates to (4π/3) times the dot product of the two constant vectors. This is valid for states with orbital angular momentum l=0, such as the ground state, because the wave function then only depends on the radial coordinate. The integral is expanded, and many terms are shown to be zero due to the integration limits of cosine and sine functions. The remaining non-zero terms are integrated, leading to the desired result.
Using the result from the previous part, we demonstrate that a specific expectation value within the hyperfine Hamiltonian, which involves dot products of electron and proton spin with the radial unit vector, is zero. This simplifies the hyperfine energy calculation significantly. This relies on separating the radial and angular parts of the integral and utilizing the fact that the angular integral evaluates to zero when combined with other terms.
With the simplification of the Hamiltonian, the hyperfine energy shift is now expressed in terms of the expectation value of the dot product of the electron and proton spin operators, multiplied by the squared wave function at the origin. The expectation value of the spin dot product is determined by considering the total spin quantum number 's', which can be 0 (anti-parallel spins) or 1 (parallel spins). This leads to two possible energy shifts, one positive and one negative, representing the hyperfine splitting, also known as spin-spin splitting.