Angular Momentum in Quantum Mechanics

Introduction

Angular momentum is the quantity that keeps a spinning top upright and a planet in orbit. In chemistry it controls the shapes of orbitals, the rules of atomic spectroscopy, the magnetic properties of atoms and the rotational spectra of molecules. Quantum mechanics treats it very differently from classical mechanics: its magnitude and only one of its components can be sharply defined, and both are quantised in units of ħ. This page builds the operators and states their eigenvalues.

Core explanation

Classical starting point. Classically, angular momentum is the vector L = r × p. Its z-component is L z = xp y − yp x, with similar expressions for L x and L y.

Quantum operators. Replacing positions and momenta by their operators (p̂ x = −iħ ∂/∂x and so on) gives, for example, L̂ z = −iħ(x ∂/∂y − y ∂/∂x). In spherical polar coordinates this simplifies beautifully to

L̂ z = −iħ ∂/∂φ

which is the operator we met for the particle on a ring. The square of the angular momentum, L̂² = L̂ x² + L̂ y² + L̂ z², depends only on the angles θ and φ:

L̂² = −ħ²[(1/sin θ) ∂/∂θ (sin θ ∂/∂θ) + (1/sin²θ) ∂²/∂φ²]

The bracketed angular operator is called the legendrian, and it is exactly the angular part of the Laplacian ∇².

Eigenvalues. Solving the eigenvalue equations with the requirement that the wavefunction be single-valued and finite on a sphere gives

L̂² eigenvalue: l(l + 1)ħ², with l = 0, 1, 2, … L̂ z eigenvalue: mₗħ, with mₗ = −l, −l + 1, …, +l

The magnitude of the angular momentum is therefore √(l(l + 1))ħ, and for each l there are 2l + 1 allowed z-components.

Commutators decide what can be known. The components do not commute with one another: [L̂ x, L̂ y] = iħL̂ z, and cyclic permutations. Non-commuting observables cannot simultaneously have definite values, so if L z is known exactly, L x and L y are uncertain. However, L̂² commutes with every component: [L̂², L̂ z] = 0. We can therefore specify the magnitude and one component together, and by convention we choose the z-component.

Why the maximum component is less than the magnitude. The largest z-component is lħ, while the magnitude is √(l(l + 1))ħ, which is always larger when l > 0. If L z equalled the magnitude, L x and L y would both be exactly zero, and we would know all three components, contradicting the commutation relation. The vector can never point straight along the z-axis.

Magnetic moments. A circulating electron is a current loop, so orbital angular momentum produces a magnetic moment proportional to L. Its z-component is −mₗμ B, where the Bohr magneton μ B = eħ/(2m e) ≈ 9.27 × 10⁻²⁴ J T⁻¹. This is why mₗ is called the magnetic quantum number.

Formulae

L̂ z = −iħ ∂/∂φ; L² = l(l + 1)ħ²; L = √(l(l + 1))ħ; L z = mₗħ; [L̂ x, L̂ y] = iħL̂ z; [L̂², L̂ z] = 0; cos θ = mₗ/√(l(l + 1)).

Step-by-step reasoning

To describe the angular momentum of a state with quantum number l:

1. Calculate the magnitude √(l(l + 1))ħ. 2. List the allowed mₗ values from −l to +l; there are 2l + 1 of them. 3. Multiply each by ħ to get the allowed z-components. 4. For the vector model, find each cone angle from cos θ = mₗ/√(l(l + 1)).

Visual explanation

Draw a vertical z-axis and a set of cones opening upwards and downwards from the origin. For l = 1 there are three: an upward cone for mₗ = +1, a flat disc for mₗ = 0 and a downward cone for mₗ = −1. The angular momentum vector, of length √2 ħ, lies somewhere on the surface of one cone, its azimuthal direction completely undetermined.

Real-world analogy

A spinning coin wobbling on a table precesses: its tilt is steady, but the direction it leans sweeps round and round. The vector model is similar: the tilt of the angular momentum relative to the z-axis is fixed, but the direction around the axis is unknown, so the x and y components are smeared out.

Real-world example

In the Zeeman effect, placing atoms in a magnetic field splits a spectral line into several components. A level with l = 1 splits into three, because the three mₗ values have different magnetic energies. Astronomers use Zeeman splitting in the light of sunspots to measure magnetic fields of a few tenths of a tesla on the Sun.

Why?

Why can only one component be known? Measuring L z involves the angle φ about the z-axis, while L x and L y involve rotations about other axes. The operators for rotation about different axes do not commute, so, like position and momentum, they obey an uncertainty relation and cannot all be sharp at once.

Common misconception

"For a p electron with mₗ = +1 the angular momentum points along z." Its z-component is ħ but its magnitude is √2 ħ, so the vector is tilted at 45° to the axis and has a non-zero, undetermined perpendicular part.

Worked example

Question: For a d electron (l = 2), calculate the magnitude of the orbital angular momentum and the smallest angle its vector can make with the z-axis.

Reasoning: Magnitude = √(2 × 3)ħ = √6 ħ = 2.449 × 1.055 × 10⁻³⁴ J s = 2.58 × 10⁻³⁴ J s. The smallest angle occurs for mₗ = +2: cos θ = 2/√6 = 0.816, so θ = 35.3°.

Answer: 2.58 × 10⁻³⁴ J s; 35.3°.

Quick check

1. How many different z-components of angular momentum are allowed for a state with l = 3, and what is the largest? Answer: Seven components are allowed (mₗ = −3 to +3), and the largest is 3ħ.

Exam focus

Learn the two eigenvalue results, l(l + 1)ħ² and mₗħ, and the rule that mₗ runs from −l to +l. Questions often ask why only L² and L z are specified together; answer with the commutation relations. Take care not to give the magnitude as lħ.

Advanced insight

The commutation relations alone, without any wavefunctions, are enough to derive that the eigenvalues must take the form j(j + 1)ħ² and m jħ with j an integer or a half-integer. Orbital motion allows only integers, because the wavefunction must be single-valued under a 2π rotation. Half-integer values appear for electron spin, which has no spatial wavefunction and therefore no such restriction.

Summary

Orbital angular momentum is represented by operators built from r × p; in spherical coordinates L̂ z = −iħ ∂/∂φ and L̂² is the angular part of the Laplacian. The eigenvalues are l(l + 1)ħ² for L² and mₗħ for L z, with l = 0, 1, 2, … and mₗ = −l … +l. Because the components do not commute, only the magnitude and one component can be specified, leading to the vector model and space quantisation.

Practice questions

1. Write the operator for the z-component of angular momentum in spherical coordinates. Answer: L̂ z = −iħ ∂/∂φ. 2. What is the magnitude of the orbital angular momentum of an electron in a p orbital? Answer: √(1 × 2)ħ = √2 ħ ≈ 1.49 × 10⁻³⁴ J s. 3. Why is it impossible for L x and L z to have definite values at the same time? Answer: Their operators do not commute (their commutator is proportional to L̂ y), so they cannot share a complete set of eigenfunctions. 4. Into how many components does a level with l = 2 split in a magnetic field, according to the simple orbital model? Answer: Five, one for each value of mₗ from −2 to +2.