Hamiltonian

From Conservapedia

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 q_i are the generalised coordinates and p_i are the momenta conjugate to these coordinates, and L is the Lagrangian. For many problems the Hamiltonian is the same as the energy.

Example

For a mass m attached to a spring of spring constant k extended by a distance x, 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 coordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.