7. Command-Line Tools

7.1. Purpose

The optional tools print coefficients, coupling tables, and rotation matrices for quick inspection. They are intended for convention checks, teaching, and small reference tables. Analysis code should call the library directly.

Build them with WIGNER_BUILD_EXECUTABLES=ON. A normal top-level build and ./scripts/build.sh already enable them. Installed tools are placed in the installation prefix’s bin directory.

All angular-momentum command-line arguments use doubled notation.

7.2. Coupling and 3j Tools

print_cg prints Clebsch-Gordan coefficients for two input angular momenta and either one specified or every allowed coupled momentum:

print_cg <two_j1> <two_j2> [two_j]

print_cg_table scans canonical rows for a selected output momentum:

print_cg_table <target_two_j> [--two-j-max=N] [--all-rows]

print_3j prints the magnetic-projection table for a fixed angular-momentum triple:

print_3j <two_j1> <two_j2> <two_j3>

print_3j_table performs a broader canonical scan ending at the requested third angular momentum:

print_3j_table <target_two_j3> [--two-j-max=N] [--all-rows]

The --all-rows option retains rows whose coefficients vanish; by default, table output focuses on nonzero content.

7.3. 6j and 9j Tools

print_6j and print_9j evaluate one symbol in the standard displayed layout:

print_6j <two_a> <two_b> <two_c> <two_d> <two_e> <two_f>
print_9j <two_a> <two_b> <two_c> <two_d> <two_e> <two_f> \
         <two_g> <two_h> <two_i>

print_6j_table and print_9j_table generate small canonical tables. The default maximum doubled angular momentum is six and can be changed with --two-j-max=N or a positional integer:

print_6j_table --two-j-max=4
print_9j_table 4

Canonical scans grow rapidly. Keep the requested maximum modest, particularly for 9j output.

7.4. Recoupling Tool

print_recoupling prints admissible intermediate momenta and coefficients for one of the supported three-angular-momentum coupling-scheme changes:

print_recoupling <two_j1> <two_j2> <two_j3> [two_J] \
    [--matrix] [--symbolic] [--scheme=12-23|12-13|13-23]

Without two_J, the tool visits all allowed totals. --matrix prints the complete transformation matrix. --symbolic supplements decimals with simple square-root forms when recognized.

7.5. Rotations and Triangle Coefficients

print_wigner_d_matrix prints a reduced rotation matrix for a fixed j:

print_wigner_d_matrix <two_j> [--symbolic]

print_triangle_table prints nonzero triangle coefficients up to a selected maximum:

print_triangle_table [--two-j-max=N]

7.6. Reading Output Safely

Tool output is presentation-oriented, not a stable machine-readable file format. Scripts that need coefficients should link wigner::wigner and write a format with an explicit schema. This avoids coupling an analysis to column widths or symbolic-printing heuristics.