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

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

where $q_{i}$ are the generalised coordinates and $p_{i}$ are the canonically conjugate momenta for these coordinates, and $L$ is the Lagrangian. The canonically conjugate momentum can be found as:

$p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}$

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

Hamilton's 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 $x$. Therefore the Lagrangian is $L={\frac {1}{2}}m{\dot {x}}^{2}-{\frac {1}{2}}kx^{2}$

The canonically conjugate momentum is

$p={\frac {\partial L}{\partial {\dot {x}}}}=m{\dot {x}}$

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 same as if we had used 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 non-relativistic Hamiltonian is

${\hat {H}}={\frac {{\hat {p}}^{2}}{2m}}+V$

with ${\hat {p}}$ being the momentum operator, $m$ the mass and $V$ the potential.

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