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(q_i, p_i) =\sum_i p_i \dot{q_i} - L </math>

where <math>q_i</math> are the generalised coordinates and <math>p_i</math> are the canonically conjugate momenta for these coordinates, and <math>L</math> is the Lagrangian. The canonically conjugate momentum can be found as:

<math>p_i = \frac{\partial L}{\partial \dot{q_i}}</math>

For many problems the Hamiltonian is the same as the total energy of the system.

Hamilton's equations are:

<math>\dot p_i = -\frac{\partial H}{\partial q_i}</math>

<math>\dot q_i =~~\frac{\partial H}{\partial p_i}.</math>

Example

For a mass <math>m</math> attached to a spring of spring constant <math>k</math> extended by a distance <math>x</math>. Therefore the Lagrangian is <math>L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} kx^2</math>

The canonically conjugate momentum is

<math>p = \frac{\partial L}{\partial \dot{x}} = m \dot{x}</math>

and so

<math> H = p \dot{x} - L </math>

<math> H = \frac{p^2}{2m}+\frac{kx^2}{2} </math>

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

The equations of motion are:

<math>\dot p = -\frac{\partial H}{\partial x} = -kx</math>

<math>\dot x =~~\frac{\partial H}{\partial p_i} = p/m</math>.

Inserting <math>p = m \dot x</math> this into the first equation, we get <math>m \ddot x = -kx</math>. This is same as if we had used Newton's second law, <math>F = ma</math>.

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 non-relativistic Hamiltonian is

<math>\hat{H} = \frac{\hat{p}^2}{2m} + V</math>

with <math>\hat{p}</math> being the momentum operator, <math>m</math> the mass and <math>V</math> the potential.

The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.