Difference between revisions of "Hamiltonian"
(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>...) |
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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 == | ||
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<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 | + | 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> | ||
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</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 | + | |
| + | 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]] | ||
Revision as of 09:46, November 16, 2008
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 coordinates and <math>p_i</math> are the momenta conjugate to these coordinates, 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 coordinates and momenta to operators. The quantum mechanical Hamiltonian is of central importance to the Schrodinger equation.