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The '''Hamiltonian''' is a quantity of great importance in both classical and quantum mechanics.  

The '''Hamiltonian''' is a quantity of great importance in both [[classical mechanics|classical]] and [[quantum mechanics]].  

== Classical mechanics ==

== Classical mechanics ==

<math>H=\sum_i p_i \dot{q_i} - L </math>

<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.

where <math>q_i</math> are the generalised coordinates and <math>p_i</math> are the [[momentum|momenta]] conjugate to these coordinates, and <math>L</math> is the [[Lagrangian]]. For many problems the Hamiltonian is the same as the energy.

===Example===

===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

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>

<math>

</math>

</math>

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

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

 

== Quantum mechanics ==

== 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]].

 

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]].

[[Category:Physics]]

[[Category:Physics]]