Notes 5: Ground state hyperfine structure Stimulated by the discovery of extraterrestrial radio emission, Hendrik van der Hulst predicted that atom hydrogen could produce radiation at a wavelength of 21 cm due to a difference in energy levels caused by the interaction of the electron and proton magnetic moments. This 21 cm line was discovered in 1951, and has proven to be very useful in mapping the distribution of atomic hydrogen in the Milky Way and other galaxies. Hamiltonian for the interaction of two magnetic moments Earlier, we related the magnetic moment of the electron to its spin by treating it as a current loop. Doing the same for the proton, we can write the proton and electron magnetic moments as gep gee μp= Sμ pe,.= − S e (5.1) 22mmpe Because of its much larger mass, the magnitude of the magnetic moment of the proton is much less than that of the electron. By treating the proton as a spinning charged spherical shell and letting its radius go to zero, it can be shown that the magnetic field due to the proton’s magnetic field in a reference frame centered on the proton is µµ 002 3 B =3.(μpp ⋅− rrˆˆ) μ + μ pδ ( r) (5.2) 43π r3 The delta function term arises from the field inside the spherical shell1. The Hamiltonian describing the interaction of the two magnetic moments is µµ 13002 Hhf =−3(μp ⋅− rrˆˆ) μp + μpδ ( r) ⋅ μe 43π r3 (5.3) µµ 002 3 =−3.(μrμrp ⋅ˆˆ)( e ⋅−) μμ pe ⋅ − μμ pe ⋅δ ( r) 43π r3 The perturbations to the energy levels due this term are referred to as hyperfine structure. 1 Interestingly, the delta function term would have a coefficient of opposite sign if we treated the magnetic dipole as two closely spaced magnetic monopoles. 1 Here we will only consider hyperfine structure in the ground state of atomic hydrogen. Since l = 0, the 1 unperturbed wave functions are spherically symmetric. The expectation value of the first term in Hhf is then zero. The expectation value of the remaining term is 2 1 µ0 gpe ge 2 Ehf = SSp⋅ e ψ100 (0.) (5.4) 6mmpe In deriving the fine structure corrections, we found that S and L did not commute with the fine structure Hamiltonian but the total angular momentum J = S + L did. Similarly Sp and Se do not commute with SSpe⋅ , but the total spin S = Sp + Se does. We find 2 132 22 SSpe⋅ =(S − Sp − S e) = ss( +−1.) (5.5) 2 22 1 Since the total spin angular momentum quantum number can be 0 or 1, Ehf takes different values depending whether the total spin eigenstate is the singlet state (antisymmetric under interchange of spins) or one of the triplet states (symmetric under interchange of spins). We see that the triplet states have higher energy than the singlet state. A decay from a triplet state to the singlet results in emission of a photon of energy 22 22 µµg ge2 g ge 1 =00pe ψ =pe ≈×−6 Eγ 100 (0) 3 5.87 10 eV. (5.6) 66mmpe mmpe π a Other applications of hyperfine structure If a nucleus of an atom or ion has non-zero spin, it will have a magnetic moment, and if it is not prevented by the Pauli exclusion principle, a spin flip can emit a photon in the microwave (cm radio) part of the spectrum. The hyperfine structure lines from D at 96 cm and 3He+ at 3.4 cm have been used to probe big bang production of light elements. The D line has been detected in observations of the Galactic center and the 3He+ line has been detected in planetary nebulae. Hyperfine structure states of trapped ions have been used as qubits in quantum computing. 2 .