Hamiltonian

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

<math>H=\sum_i p_i \dot{q_i} - L </math>

where <math>q_i</math> are the generalised co-ordinates and <math>p_i</math> are the momenta conjugate to these co-ordinates, 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 co-ordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.