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The Hamiltonian is a quantity of great importance in both classical and quantum mechanics.

Classical mechanics

In classical dynamics, the Hamiltonian is defined to be

where are the generalised co-ordinates and are the momenta conjugate to these co-ordinates, and is the Lagrangian. For many problems the Hamiltonian is the same as the energy.


For a mass attached to a spring of spring constant extended by a distance , and so

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 co-ordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.