Problems are listed in approximate order of difficulty. A single dot (•) indicates straigh...

Problems are listed in approximate order of difficulty. A single dot (•) indicates straightforward problems involving just one main concept and sometimes requiring no more than substitution of numbers in the appropriate formula. Two dots (••) identify problems that are slightly more challenging and usually involve more than one concept. Three dots (•••) indicate problems that are distinctly more challenging, either because they are intrinsically difficult or involve lengthy calculations. Needless to say, these distinctions are hard to draw and are only approximate.

•• Because of the proton’s magnetic moment every energy level of the hydrogen atom, as calculated previously, actually consists of two levels, very close together. The proton’s magnetic moment creates a magnetic field, which means that the energy of the H atom is slightly different, depending on the relative orientations of the electron and proton moments — an effect known as hyperfine splitting. You can estimate the magnitude of the hyperfine splitting as follows: (a) The magnetic field at a distance r from a magnetic moment μ is

where μ0 = 4π × 10−7 N/A2 is the permeability of space. [For simplicity, we have given the field at a point on the axis of μ. At points off the axis the field is somewhat different, but is close enough to (16.46) for the purposes of this estimate.] Given that the magnetic moment of the proton is roughly equal to the nuclear magneton μN defined in (16.12), estimate the magnetic field at a distance from a proton. (b) Taking your answer in part (a) as an estimate for the B field “seen” by an electron in the 1s state of a hydrogen atom, show that the atom’s energy differs by roughly 10−6 eV for the cases that the proton and electron spins are parallel or antiparallel. [The energy of a magnetic moment in a B field is given by (9.10).] (c) This means that the 1s state of hydrogen is really two very closely spaced levels. Compare your rough estimate with the actual separation of these levels given that the photon emitted when a hydrogen atom makes a transition between them has λ = 21 cm. (This 21-cm radiation is used by astronomers to identify hydrogen atoms in interstellar space.)