Why is orbital angular momentum quantized in integers but spin can be half-integer?

Orbital angular momentum acts on spatial wavefunctions that must return to themselves after a full rotation, which forces integer quantum numbers; spin has no spatial wavefunction and is unconstrained by single-valuedness, so it may take half-integer values.

Can the full orbital angular-momentum vector be known at once?

No; the three components do not commute, so only the total magnitude and one chosen projection can be specified simultaneously, while the other two components remain indefinite, a direct consequence of the angular-momentum algebra.

Orbital Angular Momentum

Orbital angular momentum is the quantum version of the rotational motion of a particle about a center; its magnitude and one projection are simultaneously quantized by integer quantum numbers, and its eigenfunctions are the spherical harmonics.

Definition

Orbital angular momentum is the quantum operator corresponding to the cross product of position and momentum, whose squared magnitude and one component are simultaneously quantized with integer quantum numbers, and whose eigenfunctions are the spherical harmonics.

Scope

The topic covers the orbital angular-momentum operators built from position and momentum, their commutation relations and the resulting integer quantization of magnitude and projection, the spherical harmonics as simultaneous eigenfunctions, the role of raising and lowering operators, and the appearance of orbital angular momentum in the angular part of any central-force problem.

Core questions

How are the orbital angular-momentum operators constructed from position and momentum?
Why is orbital angular momentum restricted to integer quantum numbers?
What are the spherical harmonics and why do they describe angular wavefunctions?
How does orbital angular momentum enter central-force problems?

Key concepts

angular-momentum operators
azimuthal quantum number
magnetic quantum number
spherical harmonics
central-force problem
raising and lowering operators

Key theories

Integer quantization of orbital motion: The orbital angular-momentum operators inherit the general angular-momentum algebra, but the requirement that spatial wavefunctions be single valued under rotation restricts the magnitude and projection quantum numbers to integers, unlike intrinsic spin.
Spherical harmonics: The simultaneous eigenfunctions of the squared magnitude and one projection of orbital angular momentum are the spherical harmonics, an orthonormal set of functions on the sphere that form the angular factor of the wavefunction in every spherically symmetric problem.

Clinical relevance

Orbital angular momentum labels the shapes of atomic orbitals as s, p, d, and f, organizes the periodic table and selection rules for spectral transitions, and shapes the rotational spectra of molecules probed in chemistry and astrophysics.

History

Spherical harmonics arose in classical potential theory with Laplace and Legendre; Sommerfeld's quantization and then Schrodinger's 1926 solution of central-force problems revealed them as the natural eigenfunctions of quantized orbital angular momentum.

Key figures

Pierre-Simon Laplace
Arnold Sommerfeld
Erwin Schrodinger

Seminal works

sakurai2017
cohentannoudji2019

Frequently asked questions

Why is orbital angular momentum quantized in integers but spin can be half-integer?: Orbital angular momentum acts on spatial wavefunctions that must return to themselves after a full rotation, which forces integer quantum numbers; spin has no spatial wavefunction and is unconstrained by single-valuedness, so it may take half-integer values.
Can the full orbital angular-momentum vector be known at once?: No; the three components do not commute, so only the total magnitude and one chosen projection can be specified simultaneously, while the other two components remain indefinite, a direct consequence of the angular-momentum algebra.