Angular Momentum

Angular momentum in quantum mechanics is quantized in units of ℏ. Both orbital angular momentum L and spin S are characterized by quantum numbers ℓ and s, with z-components restricted to integer or half-integer multiples of ℏ.

Key Concepts

L² eigenvalues: ℓ(ℓ+1)ℏ²; Lz = mℓℏ
Electron spin: s=1/2, ms = ±1/2
Spin angular momentum: |S| = √(3/4)ℏ
Addition of angular momenta: |j₁-j₂| ≤ j ≤ j₁+j₂
Spin-orbit coupling splits energy levels (fine structure)

Key Equations

L² eigenvalue

\hat{L}^2 Y_\ell^{m_\ell} = \ell(\ell+1)\hbar^2 Y_\ell^{m_\ell}

Lz eigenvalue

\hat{L}_z Y_\ell^{m_\ell} = m_\ell\hbar Y_\ell^{m_\ell}

Spin magnitude

|\vec{S}| = \sqrt{s(s+1)}\,\hbar = \frac{\sqrt{3}}{2}\hbar\text{ (electron)}

Addition rule

j = |j_1 - j_2|,\ldots, j_1 + j_2

Worked Example

Example Problem

Problem

An electron has orbital quantum number ℓ=2. Find |L| in units of ℏ.

Solution

|L| = √(ℓ(ℓ+1))ℏ = √6 ℏ ≈ 2.449 ℏ.

Practice

Exercises

7 problems

1 of 7

Use the angular momentum cone model to find |L| for ℓ = 3 in units of ℏ.

The cone model shows L precessing around the z-axis for ℓ = 3. The vector magnitude |L| = √(ℓ(ℓ+1)) ℏ is shown. Enter |L| in units of ℏ.

ℓ = |L| = √12 ℏ ≈ 3.4641 ℏ

|L| = ℏ

2 of 7

For ℓ = 2, count the distinct quantized values of Lz shown in the diagram. How many are there?

The diagram shows all quantized Lz states for a given ℓ. Click each level to count them. For ℓ = 2, how many distinct mℓ values are there?

ℓ = 5 levels (2ℓ+1 = 5)

Count =

3 of 7

Find the magnitude of the electron spin angular momentum |S| in units of ℏ.

Unlock Exercise 3