Neumannlikningen

Neumannlikningen is a term used in mathematical and physical contexts to denote equations or boundary-value problems associated with the name Neumann. In Scandinavian languages it may refer to either the Neumann boundary condition in partial differential equations or the von Neumann equation in quantum mechanics; the precise meaning depends on context.

In the theory of partial differential equations, the Neumann problem prescribes the normal derivative of a function on the boundary of a domain. For a domain Ω with boundary ∂Ω, a typical formulation is Δu = f in Ω with ∂u/∂n = g on ∂Ω. Solvability requires a compatibility condition (for example, the integral of f over Ω must match the flux given by g on ∂Ω), and solutions are determined up to an arbitrary additive constant for elliptic operators like the Laplacian. Neumann conditions model insulated boundaries in heat flow, specified flux in diffusion, and certain electromagnetic problems.

In quantum mechanics, the von Neumann equation (sometimes called the quantum Liouville equation) governs the time evolution of the density operator ρ: iħ dρ/dt = [H, ρ], where H is the Hamiltonian and [ , ] the commutator. This equation preserves trace and Hermiticity of ρ and describes closed-system unitary dynamics; it also serves as a starting point for treatments of open quantum systems.

The name honors mathematicians and physicists of the Neumann family (notably Carl Neumann and John von Neumann), and usage varies by discipline and language.