Question

(a) The ground state of the hydrogen atom is split by the hyperfine interaction. Indicate the level diagram and show from first principles which state lies higher in energy.

(b) The ground state of the hydrogen molecule is split into total nuclear spin triplet and singlet states. Show from first principles which state lies higher in energy.

Answer 1

(a) The hyperfine interaction is one between the intrinsic magnetic moment µ_p of the proton nucleus and the magnetic field B_e arising from the external electron structure, and is represented by the Hamiltonian H_hf = -µ_p . B_e. For the ground state, the probability density for the electron is spherically symmetric and so B_e can be considered to be in the same direction as µ_e, the intrinsic magnetic moment of the electron. Then as

B_e is antiparallel to s_e and \(-\langle \mu_p.B_e\rangle\) has the same sign as \(\langle s_e.s_p\rangle\). Let S = s_e + s_p and consider the eigenstates of S² and S_e. We have

The hyperfine interaction causes the ground state to split into two states, S = 0 and S = 1 (respectively the singlet and triplet total spin states). As H_nf has the same sign as \(\langle s_e.s_p\rangle\), the energy of the triplet states is higher. The diagram of the energy levels of the ground state is shown in Fig.

Physically, hyperfine splitting is caused by the interaction of the intrinsic magnetic moments of the electron and the proton. For the electron the intrinsic magnetic moment is antiparallel to its spin; while for the proton

the magnetic moment is parallel to its spin. For the spin triplet, the spins of the electron and the proton are parallel, and so their magnetic moments are antiparallel. For the spin singlet, the reverse is true. If the spatial wave functions are same, the Coulomb energy between the election and proton is higher for the triplet state.

(b) For the hydrogen molecule H₂, as protons are fermions, the total wave function must be antisymmetric for interchange of the two protons. Then for the nuclear spin singlet, the rotation quantum number can only be L = 0,2,4..., where L = 0 has the lowest energy; for the spin triplet, the rotation quantum number can only be L = 1,3,5,. . . , where L = 1 has the lowest energy. As the energy difference caused by difference of L is larger than that caused by difference of nuclear spins, the energy of the state L = 1 (total nuclear spin S = 1) is higher than that of the state L = 0 (total nuclear spin S = 0). So for the ground state splitting of Hz, the nuclear spin triplet (S = 1) has the higher energy.

Because the spatial wave functions of L = 1 and L = 0 states are antisymmetric and symmetric respectively, the probability for the protons to come close is larger in the latter case than in the former, and so the Coulomb interaction energy is higher (for the same principal quantum number n). However, the difference between the energies of L = 1 and L = 0 is larger for the rotational energy levels than for the Coulomb energy levels. So for the ground state splitting of hydrogen atom, the nuclear spin triplet (S = 1) has the higher energy