Spin-Orbit Coupling in Quantum Mechanics

Question: Say an electron in an atom (e.g. hydrogen) has zero orbital angular momentum (l= 0) does that mean it has zero total angular momentum?

NO, because in quantum mechanics we find that for every electron in a given atom we have to process two kinds of angular momentum: orbital (L) and spin (S). Even if the electron experiences no precession or torques it must still exhibit a total angular momentum.

Earlier, we saw for the total orbital angular momentum:

L = [l(l + 1)]^1/2 (h/2pi)


We also saw that each electron has a spin angular momentum:

S = [s(s + 1)]^1/2 (h/2pi)

for which s assumes one or other of the electron spin quantum numbers, m_s = +1/2 or (-1/2).

It can be shown that by inspection that S is always [3/4]^1/2 (h/2pi)

(and I leave it to readers to easily verify that!)

Now, we must reckon in what we call the total angular momentum or J, such that:

J = [j(j + 1)]^1/2 (h/2pi)

and j = l + s (note the common letters apply to different quanitties than the capital ones!)

Thus, for an electron with zero orbital angular momentum (l=0) we have:

j = l + s = 0 + 1/2 or l + s = 0 - 1/2

so: j = +1/2 or -1/2

Then we have:

J = [j(j + 1)]^1/2 (h/2pi) = [3/4]^1/2 (h/2pi)

for either j (which readers can also verify)

Now, we can also find (as we did with L for L(z), the projection of the total angular momentum quantum number on the z-axis (J(z)) which will be:

J(z) = m_j(h/2pi)

whwere m_j = -j, -j+1, ......+j

Readers with an intuitive grasp of vectors, or if they've worked with vectors - say in high school or college physics, will quickly see that the name of the game is to obtain a vector sum such that:

V = V(1) + V(2)

In this case, L plays the role of V(1), and S plays the role of V(2)

Then we obtain for J:

J = L + S

As any physics student knows, the way to obtain the vector sum is via the law of cosines and this is demonstrated in the diagram with the computations. This is for the case:

L = 3

S = 1/2

J = 5/2

and the angle (theta) can also be obtained (as shown).

Note that in a weak magnetic in which the atom is situated, the L-S coupled system depends on j, in other words this very angle between the vectors L and S. Meanwhile, the orientation of the atom on the whole, depends on m_j.

Problem:

State which values of l, s, and j would apply to the case in the diagram, such that the assorted angular momentum vectors have the values identified.

Next time, we will also look at how L-S coupling helps us to examine in detail atomic spectra in conjunction with energy levels in the atom.