bei48482_FM

248 Chapter Seven

The potential energy Umof a magnetic dipole of moment in a magnetic field B

is, as we know,

UmBcos  (6.38)

where is the angle between and B. The quantity cos is the component of 

parallel to B. In the case of the spin magnetic moment of the electron this component

is sz B. Hence

 cos     B

and so

Spin-orbit coupling Um BB (7.15)

Depending on the orientation of its spin vector S,the energy of an atomic electron will

be higher or lower by BBthan its energy without spin-orbit coupling. The result is

that every quantum state (except sstates in which there is no orbital angular momen-

tum) is split into two substates.

The assignment of s^12 is the only one that agrees with the observed fine-structure

doubling. Because what would be single states without spin are in fact twin states, the

2 s1 possible orientations of the spin vector Smust total 2. With 2s 1 2, the

result is s^12 .

Example 7.3

Estimate the magnetic energy Umfor an electron in the 2pstate of a hydrogen atom using the

Bohr model, whose n2 state corresponds to the 2pstate.

Solution

A circular wire loop of radius rthat carries the current Ihas a magnetic field at its center of

magnitude

B

 0 I



2 r

+ Ze – e

B

(a)(b)

Figure 7.13(a) An electron circles an atomic nucleus, as viewed from the frame of reference of the

nucleus. (b) From the electron’s frame of reference, the nucleus is circling it. The magnetic field the

electron experiences as a result is directed upward from the plane of the orbit. The interaction between

the electron’s spin magnetic moment and this magnetic field leads to the phenomenon of spin-orbit

coupling.

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