Angular Momentum and Spin
Angular momentum in quantum mechanics is governed by a universal operator algebra that quantizes both the orbital motion of particles and the intrinsic spin they carry, and combining these momenta explains atomic structure, spectra, and magnetism.
Definition
Quantum angular momentum is any set of three operators obeying the canonical angular-momentum commutation relations, whose simultaneous eigenstates of total magnitude and one projection are quantized; it encompasses orbital angular momentum, intrinsic spin, and their combinations.
Scope
The area covers the commutation relations defining quantum angular momentum, the quantization of magnitude and projection, spherical harmonics for orbital motion, intrinsic spin and the special case of spin one-half, the coupling of two or more angular momenta with Clebsch-Gordan coefficients, and the exact solution of the hydrogen atom that ties these ideas to real spectra.
Sub-topics
Core questions
- What algebraic relations define angular momentum in quantum mechanics?
- Why are both the magnitude and the projection of angular momentum quantized?
- What is spin and how does it differ from orbital angular momentum?
- How do separate angular momenta combine into a total angular momentum?
Key concepts
- commutation relations
- raising and lowering operators
- spherical harmonics
- spin one-half
- Clebsch-Gordan coefficients
- total angular momentum
Key theories
- Angular-momentum algebra
- The three components of any angular momentum satisfy fixed commutation relations from which raising and lowering operators build a ladder of states, fixing the allowed eigenvalues of total magnitude and projection to integer or half-integer multiples of the fundamental quantum.
- Spin and the addition of angular momenta
- Intrinsic spin, with no spatial wavefunction, obeys the same algebra and admits half-integer values; combining two angular momenta produces a total whose allowed values range between their sum and difference, with Clebsch-Gordan coefficients giving the change of basis.
Clinical relevance
Angular momentum and spin underlie the structure of the periodic table, the fine and hyperfine splitting of spectral lines, and magnetic phenomena; spin is the basis of nuclear magnetic resonance and magnetic resonance imaging, electron spin resonance, and spin-based qubits in quantum computing.
History
The Stern-Gerlach experiment of 1922 revealed space quantization; Goudsmit and Uhlenbeck proposed electron spin in 1925, Pauli formalized it with his spin matrices, and Wigner and others developed the group-theoretic theory of angular-momentum coupling that organizes atomic and nuclear spectra.
Key figures
- Wolfgang Pauli
- Samuel Goudsmit
- George Uhlenbeck
- Eugene Wigner
Related topics
Seminal works
- sakurai2017
- edmonds1957
Frequently asked questions
- Why can angular momentum take half-integer values?
- The angular-momentum algebra alone permits both integer and half-integer eigenvalues; orbital motion is restricted to integers by the single-valuedness of spatial wavefunctions, but intrinsic spin has no such constraint and can be half-integer, as for the electron.
- How is spin different from a spinning ball?
- Spin is an intrinsic, purely quantum form of angular momentum with no associated spatial rotation or size; treating the electron as a literal spinning sphere gives the wrong magnitude and is incompatible with relativity, so spin must be regarded as a fundamental property.