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Line 1: − The '''Hamiltonian''' is a quantity of great importance in both [[classical mechanics|classical]] and [[quantum mechanics]]. + − − == Classical mechanics == − In classical dynamics, the Hamiltonian is defined to be − − <math>H=\sum_i p_i \dot{q_i} - L </math> − − where <math>q_i</math> are the generalised coordinates and <math>p_i</math> are the [[momentum|momenta]] conjugate to these coordinates, and <math>L</math> is the [[Lagrangian]]. For many problems the Hamiltonian is the same as the energy. − − ===Example=== − For a [[mass]] <math>m</math> attached to a [[spring]] of [[spring constant]] <math>k</math> extended by a distance <math>x</math>, <math>L=m\dot{x}^2/2-kx^2/2</math> and so − − <math> − H=m\dot{x}^2-L − </math> − − <math> − H=\frac{m\dot{x}}{2}+\frac{kx^2}{2} − </math> − − which is the familiar expression for the energy of a simple [[harmonic oscillator]]. − − == Quantum mechanics == − − The Hamiltonian for many quantum mechanical systems can be obtained by writing down a corresponding classical Hamiltonian and promoting all of the coordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]]. − − [[Category:Physics]]
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