Difference between revisions of "Hamiltonian"

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(Classical mechanics)
(General tidy up)
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In classical dynamics, the Hamiltonian is defined to be
 
In classical dynamics, the Hamiltonian is defined to be
  
<math>H=\sum_i p_i \dot{q_i} - L </math>
+
<math>H(q_i, p_i) =\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 [[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 canonically conjugate [[momentum|momenta]] for these coordinates, and <math>L</math> is the [[Lagrangian]]. The canonically conjugate momentum can be found as:
  
The Hamilton equations are:
+
<math>p_i = \frac{\partial L}{\partial \dot{q_i}}</math>
 +
 
 +
For many problems the Hamiltonian is the same as the total [[energy]] of the system.
 +
 
 +
Hamilton's equations are:
  
 
:<math>\dot p_i = -\frac{\partial H}{\partial q_i}</math>
 
:<math>\dot p_i = -\frac{\partial H}{\partial q_i}</math>
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===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>. The momentum <math>p = \frac{\partial L}{\partial \dot x} = m \dot x</math> and so
+
For a [[mass]] <math>m</math> attached to a [[Hooke's Law|spring]] of spring constant <math>k</math> extended by a distance <math>x</math>. Therefore the [[Lagrangian]] is
 +
<math>L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} kx^2</math>
 +
 
 +
The canonically conjugate momentum is
 +
 
 +
<math>p = \frac{\partial L}{\partial \dot{x}} = m \dot{x}</math>
 +
 
 +
and so
  
 
<math>
 
<math>
H = p\dot{x} - L
+
H = p \dot{x} - L
 
</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.
  
 
The equations of motion are:
 
The equations of motion are:
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:<math>\dot x =~~\frac{\partial H}{\partial p_i} = p/m</math>.
 
:<math>\dot x =~~\frac{\partial H}{\partial p_i} = p/m</math>.
  
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.
+
Inserting <math>p = m \dot x</math> this into the first equation, we get <math>m \ddot x = -kx</math>. This is same as if we had used Newton's second law, <math>F = ma</math>.
  
 
== 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 coordinates 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 non-relativistic Hamiltonian is
 +
 
 +
<math>\hat{H} = \frac{\hat{p}^2}{2m} + V</math>
 +
 
 +
with <math>\hat{p}</math> being the [[momentum (physics)|momentum]] operator, <math>m</math> the [[mass (science)|mass]] and <math>V</math> the potential.
 +
 
 +
The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].
  
 +
[[Category:Mechanics]]
 
[[Category:Physics]]
 
[[Category:Physics]]

Revision as of 14:59, 16 December 2016

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 canonically conjugate momenta for these coordinates, and is the Lagrangian. The canonically conjugate momentum can be found as:

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

Hamilton's equations are:

Example

For a mass attached to a spring of spring constant extended by a distance . Therefore the Lagrangian is

The canonically conjugate momentum is

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 same as if we had used Newton's second law, .

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

with being the momentum operator, the mass and the potential.

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