4
$\begingroup$

I work with complicated angular momentum calculations related to atomic physics; nevertheless, I never need to use anything related to a phase convention (apparently because it's taken care of in a consistent way everywhere with me not noticing it), but I keep reading this term within subjects related to Clebsch-Gordan coefficients, Wigner D-matrices, 3j and 6j symbols, and Racah W-Coefficients, and more.

Can someone please explain what a phase convention is and how to use/utilize it?

Or guide me to some good book that would explain that?

$\endgroup$
1
$\begingroup$

Here's a nice couple of paragraphs from Bohr & Mottelson:

Since time reversal $T$ anticommutes with the total angular momentum, it is convenient to combine $T$ with a rotation $R$ through the angle $\pi$ about an axis perpendicular to the $z$-axis (the axis of space quantization). Such a rotation also inverts [angular momentum] $I_z$ and thus \begin{align*} [ RT, I_z ] &= 0 \\ [ RT, (\mathbf I)^2] &= 0. \end{align*} It is therefore possible to construct a set of basis states with quantum numbers $IM$, which are also eigenvectors of $RT$. By suitably choosing the phases of these states, the eignvalues of $RT$ may be set equal to unity, $$ RT \left| \alpha IM\right> = \left| \alpha IM \right>, $$ where $\alpha$ represents a set of additional quantum numbers specifying the internal structure of the states. The conventional phasing corresponds to choosing the rotation axis of $R$ to be the $y$ axis.

and later

In the representation in which $j_z$ is diagonal, the nonvanishing matrix elements of the angular momentum operators are \begin{align*} \left< jm \middle| j_z \middle| jm\right> &= m \\ \left< jm\pm1 \middle| j_x\pm ij_y \middle| jm\right> &= \sqrt{(j\mp m)(j\pm m+1)} \end{align*} The nondiagonal matrix elements of $j_x\pm ij_y$ involve arbitrary phase factors associated with the choice of relative phases for the states with different $m$. The phase convention [above] implies that the matrix elements of $j_x$ are real while those of $j_y$ are purely imaginary, since $j_x$ commutes with $RT$ while $j_y$ anticommutes with $RT$. We are thus left with arbitrary real phase factors (±1), which are conventionally fixed by the requirement that the matrix elements of $j_x ± i j_y$ be positive (Condon and Shortley, 1935).

Nowadays these phase "choices" are swallowed up in the standard definitions of the spherical harmonics (particularly the relationship between phase and $m$) and in the typical representation for the three spin axes (one diagonal, one purely real and one purely imaginary).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.