Angular Momentum

Angular momentum in quantum mechanics is quantized — both spin and orbital angular momentum are built from the same algebra.

Spin 1/2 particles

S_z = \frac{\hbar}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}

S_y = \frac{\hbar}{2}\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}

S_x = \frac{\hbar}{2} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

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The $j_1 = 1/2,\ j_2 = 1/2$ Clebsch-Gordan Coefficients
$m_1$	$m_2$	$+1/2$	$-1/2$	Uncoupled	Coupled
$+1/2$	$+1/2$	$1$	$0$	$\ket{1/2, 1/2}\ket{1/2, 1/2}$	$\ket{1, 1}$
$+1/2$	$-1/2$	$\sqrt{1/2}$	$\sqrt{1/2}$	$\ket{1/2, 1/2}\ket{1/2, -1/2}$	$\ket{1, 0}$
$-1/2$	$+1/2$	$\sqrt{1/2}$	$-\sqrt{1/2}$	$\ket{1/2, -1/2}\ket{1/2, 1/2}$	$\ket{0, 0}$
$-1/2$	$-1/2$	$0$	$1$	$\ket{1/2, -1/2}\ket{1/2, -1/2}$	$\ket{1, -1}$

The energy eigenvalues of the Hydrogen atom are given by

E_n = - \frac{1}{2n^2} \Bigg(\frac{Ze^2}{4 \pi \epsilon_0}\Bigg)^2 \frac{\mu}{\hbar^2} \quad n = 1, 2, 3, ...

Practice

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