# 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.

The Hamilton equations are:

### Example

For a mass *m* attached to a spring of spring constant *k* extended by a distance *x*, . The momentum and so

which is the familiar expression for the energy of a simple harmonic oscillator.

The equations of motion are:

- .

Inserting this into the first equation, we get . This is just Newton's second law, F = ma.

## 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.