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New page: 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>...
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=\sum_i p_i \dot{q_i} - L </math>
where <math>q_i</math> are the generalised co-ordinates and <math>p_i</math> are the [[momentum|momenta]] conjugate to these co-ordinates, and <math>L</math> is the [[Lagrangian]]. For many problems the Hamiltonian is the same as the energy.
===Example===
For a mass <math>m</math> attached to a spring of spring constant <math>k</math> extended by a distance <math>x</math>, <math>L=m\dot{x}^2/2-kx^2/2</math> and so
<math>
H=m\dot{x}^2-L
</math>
<math>
H=\frac{m\dot{x}}{2}+\frac{kx^2}{2}
</math>
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]].
[[Category:Physics]]
== Classical mechanics ==
In classical dynamics, the Hamiltonian is defined to be
<math>H=\sum_i p_i \dot{q_i} - L </math>
where <math>q_i</math> are the generalised co-ordinates and <math>p_i</math> are the [[momentum|momenta]] conjugate to these co-ordinates, and <math>L</math> is the [[Lagrangian]]. For many problems the Hamiltonian is the same as the energy.
===Example===
For a mass <math>m</math> attached to a spring of spring constant <math>k</math> extended by a distance <math>x</math>, <math>L=m\dot{x}^2/2-kx^2/2</math> and so
<math>
H=m\dot{x}^2-L
</math>
<math>
H=\frac{m\dot{x}}{2}+\frac{kx^2}{2}
</math>
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]].
[[Category:Physics]]