Difference between revisions of "Hamiltonian"

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(Classical mechanics)
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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.
 
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.
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 +
The Hamilton equations are:
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 +
:<math>\dot p_i = -\frac{\partial H}{\partial q_i}</math>
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:<math>\dot q_i =~~\frac{\partial H}{\partial p_i}.</math>
  
 
===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>. The momentum <math>p = \frac{\partial L}{\partial \dot x} = m \dot x</math> and so
  
 
<math>
 
<math>
H=m\dot{x}^2-L
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H = p\dot{x} - L
 
</math>
 
</math>
  
 
<math>
 
<math>
H=\frac{m\dot{x}}{2}+\frac{kx^2}{2}
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H = \frac{p^2}{2m}+\frac{kx^2}{2}
 
</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]].
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The equations of motion are:
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:<math>\dot p = -\frac{\partial H}{\partial x} = -kx</math>
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:<math>\dot x =~~\frac{\partial H}{\partial p_i} = p/m</math>.
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Inserting <math>p = m \dot x</math> this into the first equation, we get <math>m \ddot x = -kx</math>. This is just Newton's second law, F = ma.
  
 
== Quantum mechanics ==
 
== Quantum mechanics ==

Revision as of 00:30, 19 June 2011

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

where are the generalised coordinates and are the momenta conjugate to these coordinates, and is the Lagrangian. For many problems the Hamiltonian is the same as the energy.

The Hamilton equations are:

Example

For a mass attached to a spring of spring constant extended by a distance , . The momentum and so

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

The equations of motion are:

.

Inserting this into the first equation, we get . 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.