Theorem of Liouville (Hamiltonian)

See also: Theorem of Liouville

In physics, the theorem of Liouville , named according to the mathematician Joseph Liouville, is a theorem used by the formalism hamiltionien of traditional mechanics, but also in quantum Mécanique and Physique statistics. This theorem says that the volume of the Espace of the phases is constant along the trajectories of the sytème, in other words this volume remains constant in time.

Equation of Liouville

The equation of Liouville describes the temporal evolution of the density of probability $\backslash \; rho$ in the Espace of the phases. This density of probabilté is defined by as the Probabilité so that the state of the system is represented by a point inside volume $\backslash \; Gamma$ considered.

In traditional mechanics

One uses the generalized Coordonnées $(Q,\; p)$ where $N$ is the dimension of the system. The density of probability is defined by the probability $\backslash \; rho\; (p,\; Q)\; \backslash ,\; d^Nq\; \backslash ,\; d^Np$ to meet the state of the system in infinitesimal volume $d^Nq\; \backslash ,\; d^Np$.

Losrqu' one calculates the temporal evolution this density of probability $\backslash \; rho\; (p,\; Q)$, one obtains:

$\backslash \; frac\; \{D\; \backslash \; rho\}\; \{dt\}\; =\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; rho\}\; \{\backslash \; partial\; T\}\; +\; \backslash \; sum\_\; \{i=1\}\; ^\; \{NR\}\; \backslash \; partial\; left\; \backslash \; frac\; \{\backslash \; \backslash \; rho\}\; \{\backslash \; partial\; q\_\; \{I\}\}\; \backslash \; dowry\; \{Q\}\; \_\; \{I\}\; +\; \backslash \; frac\; \{\backslash \; partial\; \backslash \; rho\}\; \{\backslash \; partial\; p\_\; \{I\}\}\; \backslash \; dowry\; \{p\}\; \_\; \{I\}\; \backslash \; right\; =\; 0$

One then uses the canonical equations of Hamilton, by replacing them in the preceding equation:

$\backslash \; dowry\; \{Q\}\; \_i\; \backslash \; =\; \backslash \; \backslash \; frac\; \{\backslash \; partial\; H\}\; \{\backslash \; partial\; p\_i\}\; \backslash \; quad;\; \backslash \; qquad\; \backslash \; dowry\; \{p\}\; \_i\; \backslash \; =\; \backslash \; -\; \backslash \; frac\; \{\backslash \; partial\; H\}\; \{\backslash \; partial\; q\_i\}$

from where:

$\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; rho\; (p,\; Q,\; T)\; =\; \backslash \; \{\backslash ,\; \backslash \; rho\; (p,\; Q,\; T),\; H\; \backslash ,\; \backslash \}\; =\; \backslash \; \{\backslash ,\; H,\; \backslash \; rho\; (p,\; Q,\; T)\; \backslash ,\; \backslash \}$

by using the hooks of Poisson.

In quantum mechanics

According to the principle of correspondence, one can quickly deduce the equation from it from Liouville in quantum Mécanique:

$\backslash \; frac\; \{1\}\; \{I\; \backslash \; hbar\}\; H,\; \backslash \; hat\; has\; (T)\; =\; \backslash \; left\; \backslash \; \{\backslash \; hat\; H,\; \backslash \; hat\; has\; \backslash \; right\; \backslash \}\; +\; O\; (\backslash \; hbar^2)$

from where one deduces:

$\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; T\}\; \backslash \; hat\; \backslash \; rho=\; \backslash \; frac\; \{I\}\; \{\backslash \; hbar\}\; H$

Here, $\backslash \; hat\; H$ is the Hamiltonian Opérateur and $\backslash \; rho$ the Matrice density. Sometimes this equation is also named the equation of Von Neumann.

Theorem of Liouville

Equation of Liouville, one can prove the theorem of Liouville, which can be formulated like:

where well:

One can show that volume $\backslash \; constant\; Gamma$est:

$\backslash \; frac\; \{D\; \backslash \; Gamma\}\; \{dt\}\; =\; 0$

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