Hamiltonian

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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 qi are the generalised coordinates and pi 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:

\dot p_i = -\frac{\partial H}{\partial q_i}
\dot q_i =~~\frac{\partial H}{\partial p_i}.

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. The momentum p = \frac{\partial L}{\partial \dot x} = m \dot x and so


H = p\dot{x} - L


H = \frac{p^2}{2m}+\frac{kx^2}{2}

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

The equations of motion are:

\dot p = -\frac{\partial H}{\partial x} = -kx
\dot x =~~\frac{\partial H}{\partial p_i} = p/m.

Inserting p = m \dot x this into the first equation, we get m \ddot x = -kx. 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.

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