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

$H=\sum _{i}p_{i}{\dot {q_{i}}}-L$

where $q_{i}$ are the generalised co-ordinates and $p_{i}$ are the momenta conjugate to these co-ordinates, 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 $x$, $L=m{\dot {x}}^{2}/2-kx^{2}/2$ and so

$H=m{\dot {x}}^{2}-L$

$H={\frac {m{\dot {x}}}{2}}+{\frac {kx^{2}}{2}}$

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.