Orbital angular momentum operator

The orbital angular momentum operators are defined by $ˆ 𝐋 = ˆ 𝐫 \times ˆ 𝐩$ , or in cartesian coördinates

ˆ 𝐿 𝑥 = ˆ 𝑦 ˆ 𝑝 𝑧 - ˆ 𝑧 ˆ 𝑝 𝑦, ˆ 𝐿 𝑦 = ˆ 𝑧 ˆ 𝑝 𝑥 - ˆ 𝑥 ˆ 𝑝 𝑧, ˆ 𝐿 𝑧 = ˆ 𝑥 ˆ 𝑝 𝑦 - ˆ 𝑦 ˆ 𝑝 𝑥

correspond to the observable orbital angular momentum for a single particle in 3D position space. The total orbital angular momentum operator $ˆ 𝐿 2 = ˆ 𝐋 \cdot ˆ 𝐋$ has eigenvalues quantized by $ℓ = 0, 1, \dots$ while $ˆ 𝐿 𝑧$ ¹ is quantized by $𝑚 = - ℓ, - ℓ + 1, \dots, ℓ - 1, ℓ$ .

ˆ 𝐿 2 | ℓ, 𝑚 ⟩ = ℏ ℓ (ℓ + 1) | ℓ, 𝑚 ⟩ ˆ 𝐿 𝑧 | ℓ, 𝑚 ⟩ = ℏ 𝑚 | ℓ, 𝑚 ⟩

Spherical coördinates

In spherical coördinates, they may be expressed as²

ˆ 𝐿 𝑥 = - 𝑖 ℏ (- s i n 𝜙 𝜕 𝜕 𝜃 - c o s 𝜙 c o t 𝜃 𝜕 𝜕 𝜙) ˆ 𝐿 𝑦 = - 𝑖 ℏ (+ c o s 𝜙 𝜕 𝜕 𝜃 - s i n 𝜙 c o t 𝜃 𝜕 𝜕 𝜙) ˆ 𝐿 𝑧 = - 𝑖 ℏ 𝜕 𝜕 𝜙

Properties

The operators obey the commutation relation $[ˆ 𝐿 𝑗, ˆ 𝐿 𝑘] = 𝑖 ℏ 𝜖 𝑗 𝑘 ℓ ˆ 𝐿 ℓ$ , hence they form a Lie algebra isomorphic to 𝔰𝔬(3).
The eigenfunctions are the spherical harmonics which are related via raising and lowering operators (see Irreps of $S O (3)$ ). It follows only integer values of $ℓ$ are allowed.

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Selected without loss of generality by symmetry. ↩
2018. Introduction to quantum mechanics, §4.3.2, p. 163. ↩

Orbital angular momentum operator

Spherical coördinates

Properties

Footnotes