Angular Momentum in Quantum Mechanics
Angular momentum in classical physics is the rotational analogue of linear momentum, given by L = r × p for a particle moving around a point. In quantum mechanics, angular momentum becomes quantised: its magnitude and its component along any chosen axis can take only specific discrete values.
This quantisation explains the shapes of atomic orbitals, the splitting of spectral lines in magnetic fields, and the periodic table itself. Electrons in atoms carry two kinds: orbital angular momentum from their motion around the nucleus, and spin, an intrinsic form with no classical counterpart. Understanding orbital angular momentum is the first step toward both.
Key Concepts
- Orbital quantum number l: a non-negative integer 0, 1, 2, … for each state. l = 0, 1, 2, 3 are labelled s, p, d, f in chemistry.
- Magnetic quantum number m_l: ranges from −l to +l in integer steps, giving 2l + 1 possible orientations.
- Quantised magnitude: |L| depends only on l, never on m_l, and is slightly larger than the maximum measurable component.
- Space quantisation: only the component along one axis, usually called L_z, is measured, and it takes values m_l ħ.
Formulas
Magnitude: |L| = √[l(l + 1)] ħ, in units of J·s (kg·m²/s). For l = 0 the angular momentum is exactly zero.
z-component: L_z = m_l ħ, where m_l = −l, −l + 1, …, +l. Applied together, these rules replace the classical free rotation with a small set of allowed orientations, often pictured as vectors lying on cones around the z-axis because L_x and L_y remain uncertain.
Worked Example
An electron is in a d orbital, so l = 2. List the possible L_z values and find |L|.
- m_l takes values −2, −1, 0, +1, +2, giving five orientations.
- L_z values are therefore −2ħ, −ħ, 0, +ħ, +2ħ. Numerically 2ħ = 2.11 × 10⁻³⁴ J·s.
- |L| = √[2 × 3] ħ = √6 ħ ≈ 2.449 ħ = 2.58 × 10⁻³⁴ J·s. Note |L| exceeds the largest component 2ħ, so the vector can never point fully along z.
Common Misconceptions
- Electrons orbit like planets on definite paths. Wrong. Orbitals are probability distributions characterised by l and m_l, not trajectories.
- |L| equals l ħ. Wrong. The correct magnitude is √[l(l+1)] ħ, which is always a little larger than l ħ for l greater than zero.
- All components can be known at once. Wrong. Only |L|² and one component, conventionally L_z, can be simultaneously definite; the others stay uncertain.
Real-World Applications
Angular momentum rules govern atomic spectra, selection rules for light emission, and the Zeeman splitting used to measure magnetic fields in stars. Chemists use s, p, d and f orbital shapes from l to predict bonding geometry, and NMR and MRI rely on quantised angular momentum of nuclei.
To summarise, quantum angular momentum is fixed in magnitude by l and in one component by m_l, giving atoms their characteristic structure.