N.13 Combining angular momentum factors

Angular momenta from different sources can be combined using the Clebsch-Gordan coefficients of chapter 12.7. For example, you can combine orbital and spin angular momenta of a particle that way, or the angular momenta of different particles.

But sometimes you need to multiply angular momentum states of the same source together. For example, you may need to figure out a product of spherical harmonics, chapter 4.2.3, like

\begin{displaymath}
Y_{l_1}^{m_1}Y_{l_2}^{m_2}
\end{displaymath}

where the position coordinates refer to the same particle. If you multiply a few examples from table 4.3 together, you quickly see that the combinations are not given by the Clebsch-Gordan coefficients of figure 12.6.

One way to understand the angular momentum properties of the above product qualitatively is to consider the case that the second spherical harmonic takes the coordinates of an imagined second particle. Then you can use the normal procedures to figure out the properties of the two-particle system. And you can get the corresponding properties of the original one-particle system by restricting the two-particle wave function coordinates to the subset in which the particle coordinates are equal.

Note in doing so that angular momentum properties are directly related to the effect of coordinate system rotations, {A.19}. Coordinate system rotations maintain equality of particle coordinates; they stay within the subset. But inner products for the two-particle system will obviously be different from those for the reduced one-particle system.

And Clebsch-Gordan coefficients are in fact inner products. They are the inner product between the corresponding horizontal and vertical states in figures 12.5 and 12.6. The correct coefficients for the product above are still related to the Clebsch-Gordan ones, though you may find them in terms of the equivalent Wigner 3j coefficients.

Wigner noticed a number of problems with the Clebsch-Gordan coefficients:

1.
Clebsch and Gordan, and not Wigner, get credit for them.
2.
They are far too easy to type.
3.
They have an intuitive interpretation.
So Wigner changed the sign on one of the variables, took out a common factor, and reformatted the entire thing as a matrix. In short
\begin{displaymath}
\left(
\begin{array}{ccc}
j_1 & j_2 & j_3 \\
m_1 & ...
...t\big\vert j_1\,m_1\big\rangle \big\vert j_2\,m_2\big\rangle
\end{displaymath} (N.1)

Behold, the spanking new Wigner 3j symbol. Thus Wigner succeeded by his hard work in making physics a bit more impenetrable still than before. A big step for physics, a small step back for mankind.

The most important thing to note about this symbol/coefficient is that it is zero unless

\begin{displaymath}
m_1 + m_2 + m_3 = 0 \quad \mbox{and}\quad \vert j_1-j_2\ve...
...qslant$}}j_3 \mathrel{\raisebox{-.7pt}{$\leqslant$}}j_1 + j_2
\end{displaymath}

The right-hand conditions are the so-called triangle inequalities. The ordering of the $j$-​values does not make a difference in these inequalities. You can swap the indices 1, 2, and 3 arbitrarily around.

If all three $m$ values are zero, then the symbol is zero if the sum of the $j$ values is odd. If the sum of the $j$ values is even, the symbol is not zero unless the triangle inequalities are violated.

If you need an occasional value for such a symbol that you cannot find in figure 12.5 and 12.6 or more extensive tables elsewhere, there are convenient calculators on the web, [[16]]. There is also software available to evaluate them. Note further that {D.66} gives an explicit expression.

In literature, you may also encounter the so-called Wigner 6j” and “9j symbols. They are typically written as

\begin{displaymath}
\left\{
\begin{array}{ccc}
j_1 & j_2 & j_3 \\
l_1 &...
..._{23} \\
j_{31} & j_{32} & j_{33}
\end{array}
\right\}
\end{displaymath}

They appear in the combination of angular momenta. If you encounter one, there are again calculators on the web. The most useful thing to remember about 6j symbols is that they are zero unless each of the four triads $j_1j_2j_3$, $j_1l_2l_3$, $l_1j_2l_3$, and $l_1l_2j_3$ satisfies the triangle inequalities.

The 9j symbol changes by at most a sign under swapping of rows, or of columns, or transposing. It can be expressed in terms of a sum of 6j symbols. There are also 12j symbols, if you cannot get enough of them.

These symbols are needed to do advanced computations but these are far outside the scope of this book. And they are very abstract, definitely not something that the typical engineer would want to get involved in in the first place. All that can be done here is to mention a few key concepts. These might be enough keep you reading when you encounter them in literature. Or at least give a hint where to look for further information if necessary.

The basic idea is that it is often necessary to know how things change under rotation of the axis system. Many derivations in classical physics are much simpler if you choose your axes cleverly. However, in quantum mechanics you face problems such as the fact that angular momentum vectors are not normal vectors, but are quantized. Then the appropriate way of handling rotations is through the so-called Wigner-Eckart theorem. The above symbols then pop up in various places.

For example, they allows you to do such things as figuring out the derivatives of the harmonic polynomials ${\cal{Y}}_l^m$ of table 4.3, and to define the vector spherical harmonics $\vec{Y}_{JlM}$ that generalize the ordinary spherical harmonics to vectors. A more advanced treatment of vector bosons, {A.20}, or photon wave functions of definite angular momentum, {A.21.7}, would use these. And so would a more solid derivation of the Weisskopf and Moszkowski correction factors in {A.25.8}. You may also encounter symbols such as ${\cal{D}}_{m'm}^{(j)}$ for matrix elements of finite rotations.

All that will be done here is give the derivatives of the harmonic polynomials $r^lY_l^m$, since that formula is not readily available. Define the following complex coordinates $x_\mu$ for $\mu$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\vphantom0\raisebox{1.5pt}{$-$}$1, 0, 1:

\begin{displaymath}
\mu=-1{:}\quad x_{-1} = \frac{x-{\rm i}y}{\sqrt2} \qquad
...
...z \qquad
\mu=1{:} \quad x_{1} = - \frac{x+{\rm i}y}{\sqrt2}
\end{displaymath}

Then

\begin{displaymath}
\frac{\partial r^l Y_l^m}{\partial x_\mu}
= (-1)^{\mu+1}...
...angle r^{l-1}Y_{l-1}^{m-\mu}
= C_\mu r^{l-1}Y_{l-1}^{m-\mu}
\end{displaymath}

where the inner product of kets is a Clebsch-Gordan coefficient and

\begin{eqnarray*}
&& \displaystyle
C_{-1} = \sqrt{\frac{(2l+1)(l-m)(l-m-1)}{...
...\displaystyle
C_1 = \sqrt{\frac{(2l+1)(l+m)(l+m-1)}{2(2l-1)}}
\end{eqnarray*}

or zero if the final magnetic quantum number is out of bounds.

If just having a rough idea of what the various symbols are is not enough, you will have to read up on them in a book like [13]. There are a number of such books, but this particular book has the redeeming feature that it lists some practical results in a usable form. Some highlights: general expression for the Clebsch-Gordan coefficients on p. 45; wrong definition of the 3j coefficient on p. 46, (one of the rare mistakes; correct is above), symmetry properties of the 3j symbol on p. 47; 3j symbols with all $m$ values zero on p. 50; list of alternate notations for the Clebsch-Gordan and 3j coefficients on p.52, (yes, of course it is a long list); integral of the product of three spherical harmonics, like in the electric multipole matrix element, on p. 63; the correct expression for the product of two spherical harmonics with the same coordinates, as discussed above, on p. 63; effect of nuclear orientation on electric quadrupole moment on p. 78; wrong derivatives of spherical harmonics times radial functions on p. 69, 80, (another 5 rare mistakes in the second part of the expression alone, see above for the correct expression for the special case of harmonic polynomials); plane wave in spherical coordinates on p. 81. If you do try to read this book, note that $\gamma$ stands for other quantum numbers, as well as the Euler angle. That is used well before it is defined on p. 33. If you are not psychic, it can be distracting.