Combine spins and orbital terms with confidence. Check allowed total values and magnetic sums. Save outputs fast for lab documentation.
Choose quantum coupling for j,m rules, or classical vectors for magnitude and angle.
| Case | j1, m1 | j2, m2 | M | Allowed J |
|---|---|---|---|---|
| Spin-1/2 + Spin-1/2 | 1/2, 1/2 | 1/2, -1/2 | 0 | 0, 1 |
| Spin-1/2 + Spin-1 | 1/2, 1/2 | 1, 0 | 1/2 | 1/2, 3/2 |
| Classical example | L1=2 | L2=3 | θ=60° | L≈4.358899 |
Quantum coupling: add two angular momenta j1 and j2.
Classical vectors: resultant magnitude
L = √(L1² + L2² + 2 L1 L2 cos θ)
Angular momentum coupling appears in atomic structure, spin systems, and scattering. This calculator summarizes allowed totals and projections so you can focus on physics instead of bookkeeping.
In quantum mode it lists every permitted total J from inputs j1 and j2, then flags which totals are compatible with your magnetic sum M. It steps J in increments of 1, matching standard coupling rules. In classical mode it computes the resultant magnitude from L1, L2, and angle θ, plus the full min–max range. Exports include a timestamp and the exact inputs used.
Each j must be an integer or half-integer such as 0, 1/2, 1, 3/2. For a given j, the magnetic quantum number m has exactly 2j+1 discrete values: m = −j, −j+1, …, +j. For example, j=3/2 yields four projections: −3/2, −1/2, 1/2, 3/2. Invalid m steps are rejected to prevent silent mistakes.
The coupled total is restricted by the triangle rule: J = |j1−j2|, |j1−j2|+1, …, j1+j2. If j1=1/2 and j2=1, the only totals are J=1/2 and J=3/2. For two spins 1/2 and 1/2, the familiar totals J=0 and J=1 appear. The table shows each J explicitly so you can confirm your coupling scheme.
The projection adds linearly: M = m1 + m2. A particular J multiplet contributes only if |M| ≤ J. This is why, for large |M|, some smaller J values become impossible even though they satisfy the triangle rule.
Every J carries degeneracy 2J+1. The product basis contains (2j1+1)(2j2+1) states, while the coupled basis contains ΣJ(2J+1) states across the allowed J values; these must match. For j1=1/2 and j2=1, 2×3=6 product states equal (2)+(4)=6 coupled states. The calculator reports both counts so you can quickly sanity-check dimensionality.
For vectors, the magnitude is L = √(L1² + L2² + 2L1L2 cosθ). The extrema are Lmin=|L1−L2| at θ=180° and Lmax=L1+L2 at θ=0°. With L1=2 and L2=3 at θ=60°, the resultant is about 4.3589, which falls within [1,5].
Use quantum mode to narrow J channels before computing Clebsch–Gordan coefficients or building Hamiltonians. Use classical mode to validate measured resultants against geometric bounds. Exporting CSV/PDF preserves inputs, constraints, and tables for reproducible lab notes.
Yes. Enter values as 3/2 or decimals like 1.5. The form validates that j is integer/half-integer and that m follows the allowed step pattern for the chosen j.
When coupling j1 and j2, the total J runs from |j1−j2| to j1+j2 in integer steps. This holds whether the endpoints are integers or half-integers.
Because the projection must satisfy |M| ≤ J, where M=m1+m2. If |M| is larger than a candidate J, that multiplet cannot contain a state with your selected projections.
No. It reports kinematic constraints (triangle rule, M compatibility, degeneracies, and state counts). Use a dedicated CG/3j library if you need amplitudes or coupling coefficients.
The product basis has (2j1+1)(2j2+1) states. The coupled basis redistributes them across J multiplets with ΣJ(2J+1). Matching counts confirms your coupling range is consistent.
Any consistent unit set works (e.g., kg·m²/s, N·m·s, or arbitrary units). The formulas use magnitudes and angles, so only relative consistency matters.
The built-in single-page PDF keeps output readable, so very long tables are shortened. The CSV export includes the full table and is best for complete archival.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.