Difference between revisions of "Hamiltonian"

Latest revision as of 04:15, May 27, 2017

The Hamiltonian is a quantity of great importance in both classical and quantum mechanics. Whereas the Lagrangian treats each generalised coordinate and its rate of change as independent, the Hamiltonian treats the generalised coordinate and its canonically conjugate momentum as independent. Lagrangian mechanics produces $N$ second order differential equations, one for each generalised coordinate. Hamiltonian mechanics leads to $2N$ first order differential equations, which makes it often easier for solving problems computationally.

Classical mechanics

In classical dynamics, the Hamiltonian is defined to be

$H(q_{i},p_{i})=\sum _{i}p_{i}{\dot {q_{i}}}-L$

where $q_{i}$ are the generalised coordinates and $p_{i}$ are the canonically conjugate momenta for these coordinates, and $L$ is the Lagrangian. The canonically conjugate momentum can be found as:

$p_{i}={\frac {\partial L}{\partial {\dot {q_{i}}}}}$

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

Hamilton's equations are:

{\dot {p}}_{i}=-{\frac {\partial H}{\partial q_{i}}}

{\dot {q}}_{i}=~~{\frac {\partial H}{\partial p_{i}}}.

Example

For a mass $m$ attached to a spring of spring constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} extended by a distance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . Therefore, the Lagrangian is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} kx^2}

The canonically conjugate momentum is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = \frac{\partial L}{\partial \dot{x}} = m \dot{x}}

and so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = p \dot{x} - L }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H = \frac{p^2}{2m}+\frac{kx^2}{2} }

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

The equations of motion are:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot p = -\frac{\partial H}{\partial x} = -kx}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dot x =~~\frac{\partial H}{\partial p_i} = p/m} .

Inserting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p = m \dot x} this into the first equation, we get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \ddot x = -kx} . This is same as if we had used Newton's second law, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 non-relativistic Hamiltonian is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H} = -\frac{\hat{p}^2}{2m} + V}

with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p}} being the momentum operator, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} the mass and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} the potential. Substituting in for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{p}} gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V}

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

@@ Line 1: / Line 1: @@
-Page Corrected By Anon
+The '''Hamiltonian''' is a quantity of great importance in both [[classical mechanics|classical]] and [[quantum mechanics]]. Whereas the [[Lagrangian Dynamics|Lagrangian]] treats each generalised coordinate and its rate of change as independent, the Hamiltonian treats the generalised coordinate and its canonically conjugate momentum as independent. Lagrangian mechanics produces <math>N</math> second order differential equations, one for each generalised coordinate. Hamiltonian mechanics leads to <math>2N</math> first order differential equations, which makes it often easier for solving problems computationally.
+== Classical mechanics ==
+In classical dynamics, the Hamiltonian is defined to be
+<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 canonically conjugate [[momentum|momenta]] for these coordinates, and <math>L</math> is the [[Lagrangian]]. The canonically conjugate momentum can be found as:
+<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 q_i =~~\frac{\partial H}{\partial p_i}.</math>
+===Example===
+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>
+H = p \dot{x} - L
+</math>
+<math>
+H = \frac{p^2}{2m}+\frac{kx^2}{2}
+</math>
+which is the familiar expression for the energy of a simple harmonic oscillator.
+The equations of motion are:
+:<math>\dot p = -\frac{\partial H}{\partial x} = -kx</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 same as if we had used Newton's second law, <math>F = ma</math>.
+== 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
+<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. Substituting in for <math>\hat{p}</math> gives
+<math>\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V</math>
+The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].
+[[Category:Mechanics]]

Difference between revisions of "Hamiltonian"

Latest revision as of 04:15, May 27, 2017

Classical mechanics

Example

Quantum mechanics

Navigation menu

Search