The 21 cm Line: One of the most important windows to the mysteries of the
cosmos is the 21 cm line. With it, astronomers map hydrogen throughout the
universe. An important trait is that it involves a highly forbidden transition
that is, accordingly, quite long-lived. But it is also an excellent example of
the coupling of angular momenta. Hydrogen's ground state has no spin-orbit
interaction - for \(\boldsymbol{\theta}=0,\) there is no orbit. However, the
proton and electron magnetic moments do interact. Consider the following
simple model. (a) The proton sees itself surrounded by a spherically symmetric
cloud of Is electron, which has an intrinsic magnetic dipole moment/spin that,
of course, has a direction. For the purpose of investigating its effect on the
proton, treat this dispersed magnetic moment as behaving effectively like a
single loop of current whose radius is \(a_{0}\), then find the magnetic tield
at the middle of the loop in tenns of \(e, h, m_{e}, \mu_{0},\) and \(a_{0} .\)
(b) The proton sits right in the middle of the electron's magnetic moment.
Like the electron. the proton is a spin- \(\frac{1}{2}\) particle, with only two
possible orientations in a magnetic field. Noting, however, that its spin and
magnetic moment are parallel rather than opposite, would the interaction
energy be lower with the proton's spin aligned or antialigned with that of the
electron'? (c) For the proton, \(g_{p}\) is \(5.6 .\) Obtain a rough value for the
energy difference between the two orientations. (d) What would be the
wavelength of a photon that carries away this energy difference?
