Bohr–van Leeuwen theorem

The Bohr–van Leeuwen theorem states that classical equilibrium statistical mechanics predicts zero magnetization for matter in a magnetic field, even when charged particles interact.

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Classical partition function: A phase-space integral that assigns statistical weights to states of a classical system at thermal equilibrium. The theorem follows because its equilibrium integral can be made independent of the magnetic field.
Statistical mechanics: The framework that derives macroscopic properties from the statistical behavior of microscopic states. The theorem is a statement about what equilibrium statistical mechanics permits in the classical limit.
Landau diamagnetism: The orbital diamagnetism of conduction electrons caused by quantized cyclotron motion in a magnetic field. It is a quantum magnetic response that the classical theorem forbids.
Niels Bohr: A Danish physicist who developed an early quantum model of the atom and contributed to quantum theory. Bohr established the result in the context of explaining why classical physics could not account for magnetism.
Minimal coupling: The prescription of including electromagnetic potentials in a charged particle’s Hamiltonian through its canonical momentum. It introduces the vector potential whose shift can be absorbed into the momentum integration.
Classical mechanics: The theory describing motion through deterministic laws such as Newton’s and Hamilton’s equations. The theorem’s zero-magnetization result applies to classical, rather than quantum, equilibrium mechanics.
Pauli paramagnetism: The spin-related paramagnetism of conduction electrons arising from unequal occupation of spin states in a magnetic field. It gives metals a magnetic response through electron spin, absent from classical mechanics.
Hendrika Johanna van Leeuwen: A Dutch physicist whose 1919 doctoral work demonstrated the absence of classical equilibrium magnetism. Her dissertation gave a rigorous formulation of the theorem and its implications for classical theory.
Canonical momentum: The momentum conjugate to a coordinate in Hamiltonian mechanics, which can differ from mechanical momentum in electromagnetic fields. Shifting the integration variable in canonical momentum removes the vector potential from the classical partition function.
Lorentz force: The force on a charged particle due to electric and magnetic fields, proportional to its velocity crossed with the magnetic field. It explains why a magnetic field bends charged-particle trajectories without doing work on them.