Hamiltonianska

Hamiltonian is a term that can refer to several related concepts in mathematics and physics. In classical mechanics, a Hamiltonian is a function that describes the total energy of a system. It is typically expressed in terms of generalized coordinates and their corresponding conjugate momenta. The Hamiltonian formulation of classical mechanics provides an alternative to the Lagrangian formulation and is often more convenient for studying conserved quantities and for the transition to quantum mechanics.

In quantum mechanics, the Hamiltonian operator, denoted by H, plays a similar role. It is an operator that corresponds to the total energy of a quantum system. The time evolution of a quantum state is governed by the Schrödinger equation, which involves the Hamiltonian operator. Solving the eigenvalue problem for the Hamiltonian operator yields the possible energy levels of the system.

The term Hamiltonian can also appear in other contexts. For example, in graph theory, a Hamiltonian cycle is a cycle in a graph that visits every vertex exactly once. Similarly, a Hamiltonian path visits every vertex exactly once but does not necessarily return to the starting vertex. The problem of finding a Hamiltonian cycle or path is a well-known NP-complete problem.