9Experiment 9

Angular Momentum: The Vector Model and Quantum Numbers

Visualise the vector-cone model for orbital angular momentum, verify quantum number constraints, and apply the triangle rule for angular momentum addition.

A. Single angular momentum — vector cone model

|L|² = ℓ(ℓ+1)ħ² and Lz = mℓħ are simultaneously definite. The cone angle θ = arccos(Lz/|L|) never reaches 0° — L never points exactly along z.

ℓ =1

mℓ =1

Allowed mℓ values-1, 0, 1

|L|/ħ = √[ℓ(ℓ+1)]1.4142

Lz/ħ = mℓ1.0000

Cone half-angle θ45.00°

Degeneracy = 2ℓ+1 = 3 m-values for this ℓ.

B. Angular momentum addition (triangle rule)

j₁ =

j₂ =

Choose j₁ and j₂, then click Compute.

C. Spin-½

Click the button to compute.

Spin obeys the same algebra as orbital angular momentum (same eigenvalue equations) but with s=1/2 fixed — no spatial representation exists.

Observations

Record your measured data in the tables below. Refer to the lab manual for the full procedure.

Table 9.1 — Single angular momentum (Part A)

$ℓ$	$∣ L ∣/ ℏ$	No. of mℓ	Min cone angle

Table 9.2 — Angular momentum addition (Part B)

j₁	j₂	Allowed total j (degeneracies)

Table 9.3 — Spin-½ cone (Part C)

mₛ	Cone half-angle	\|S\|/ℏ