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The Spin-Orbit Effect

In order to further explain the structure of the hydrogen atom, one needs to consider that the electron not only has orbital angular momentum tex2html_wrap_inline1438 , but also intrinsic angular momentum tex2html_wrap_inline1440 , called spin. There is an associated spin operator tex2html_wrap_inline1440 , as well as operators tex2html_wrap_inline1444 and tex2html_wrap_inline1446 , just as with tex2html_wrap_inline1438 . Usually written in matrix form, these operators yield results analogous to tex2html_wrap_inline1358 and tex2html_wrap_inline1362 when acting on the wavefunction tex2html_wrap_inline1306 ,


where s and tex2html_wrap_inline1458 are quantum numbers defining the magnitude of the spin angular momentum and its projection onto the z-axis, respectively. For an electron s=1/2 always, and hence the electron can have tex2html_wrap_inline1464 .

Associated with this angular momentum is an intrinsic magnetic dipole moment




is a fundamental unit of magnetic moment called the Bohr magneton. The number tex2html_wrap_inline1466 is called the spin gyromagnetic ratio of the electron, expected from Dirac theory to be exactly 2 but known experimentally to be tex2html_wrap_inline1468 . This is to be compared to the magnetic dipole moment associated with the orbit of the electron,


where tex2html_wrap_inline1470 is the orbital gyromagnetic ratio of the electron. That is, the electron creates essentially twice as much dipole moment per unit spin angular momentum as it does per unit orbital angular momentum. One expects these magnetic dipoles to interact, and this interaction constitutes the spin-orbit effect.

Figure 2: On the left, an electron moves around the nucleus in a Bohr orbit. On the right, as seen by the electron, the nucleus is in a circular orbit.

The interaction is most easily analyzed in the rest frame of the electron, as shown in figure 2. The electron sees the nucleus moving around it with speed v in a circular orbit of radius r, producing a magnetic field


In terms of the electron orbital angular momentum L=mrv, the field may be written


The spin dipole of the electron has potential energy of orientation in this magnetic field given by


However, the electron is not in an inertial frame of reference. In transforming back into an inertial frame, a relativistic effect known as Thomas precession is introduced, resulting in a factor of 1/2 in the interaction energy. With this, the Hamiltonian of the spin-orbit interaction is written


With this term added to the Hamiltonian, the operators tex2html_wrap_inline1362 and tex2html_wrap_inline1446 no longer commute with the Hamiltonian, and hence the projections of tex2html_wrap_inline1438 and tex2html_wrap_inline1440 onto the z-axis are not conserved quantities. However, one can define the total angular momentum operator


It can be shown that the corresponding operators tex2html_wrap_inline1490 and tex2html_wrap_inline1492 do commute with this new Hamiltonian. Physically what happens is that the dipoles associated with the angular momentum vectors tex2html_wrap_inline1440 and tex2html_wrap_inline1438 exert equal and opposite torques on each other, and hence they couple together and precess uniformly around their sum tex2html_wrap_inline1498 in such a way that the projection of tex2html_wrap_inline1498 on z-axis remains fixed. The operators tex2html_wrap_inline1490 and tex2html_wrap_inline1492 acting on tex2html_wrap_inline1306 yield


where j has possible values


For a hydrogenic atom s=1/2, and hence the only allowed values are j=l-1/2,l+1/2, except for l=0, where only j=1/2 is possible. Figure 3 illustrates spin-orbit coupling for particular values of l, j, and tex2html_wrap_inline1524 .

Figure 3: Spin-orbit coupling for a typical case of s=1/2, l=2, j=5/2, tex2html_wrap_inline1196 , showing how tex2html_wrap_inline1438 and tex2html_wrap_inline1440 precess about tex2html_wrap_inline1498 .

Since the coupling is weak and hence the interaction energy is small relative to the principle energy splittings, it is sufficient to calculate the energy correction by first-order perturbation theory using the previously found wavefunctions. The energy correction is then


The value of tex2html_wrap_inline1540 is easily found by calculating


and hence when acting on tex2html_wrap_inline1306 ,


One then needs to calulate the expectation of tex2html_wrap_inline1544 , which is more complicated. The answer is


where the value s=1/2 has been included.

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Randy Telfer
Mon Oct 26 15:21:07 EST 1998