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# 5.2: Evolución temporal de la matriz de densidad

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La ecuación de movimiento para la matriz de densidad se deriva naturalmente de la definición$$\rho$$ y la ecuación de Schrödinger dependiente del tiempo.

\begin{align} \dfrac {\partial \rho} {\partial t} &= \dfrac {\partial} {\partial t} [ | \psi \rangle \langle \psi | ] \\[4pt] &= \left[ \dfrac {\partial} {\partial t} | \psi \rangle \right] \langle \psi | + | \psi \rangle \dfrac {\partial} {\partial t} \langle \psi | \\[4pt] &= \dfrac {- i} {\hbar} H | \psi \rangle \langle \psi | + \dfrac {i} {\hbar} | \psi \rangle \langle \psi | H . \label{4.13} \\[4pt] &= \dfrac {- i} {\hbar} [ H , \rho ] \label{4.14} \end{align}

La ecuación\ ref {4.14} es la ecuación de Liouville-Von Neumann. Es isomórfico a la ecuación de movimiento de Heisenberg, ya que también$$ρ$$ es un operador. La solución a la Ecuación\ ref {4.14} es

$\rho (t) = U \rho ( 0 ) U^{\dagger} \label{4.15}$

Esto se puede demostrar integrando primero la Ecuación\ ref {4.14} para obtener

$\rho (t) = \rho ( 0 ) - \dfrac {i} {\hbar} \int _ {0}^{t} d \tau [ H ( \tau ) , \rho ( \tau ) ] \label{4.16}$

Si expandimos la Ecuación\ ref {4.16} sustituyendo iterativamente en sí misma, la expresión es la misma que cuando sustituimos

$U = \exp _ {+} \left[ - \dfrac {i} {\hbar} \int _ {0}^{t} d \tau H ( \tau ) \right] \label{4.17}$

en Ecuación\ ref {4.15} y recoger términos por órdenes de$$H(\tau)$$.

Tenga en cuenta que la ecuación\ ref {4.15} y la invarianza cíclica de la traza implican que el valor de expectativa dependiente del tiempo de un operador puede calcularse propagando el operador (Heisenberg) o la matriz de densidad (Schrödinger o imagen de interacción):

\left.\begin{aligned} \langle \hat {A} (t) \rangle & = \operatorname {Tr} [ \hat {A} \rho (t) ] \\[4pt] & = \operatorname {Tr} \left[ \hat {A} U \rho _ {0} U^{\dagger} \right] \\[4pt] & = \operatorname {Tr} \left[ \hat {A} (t) \rho _ {0} \right] \end{aligned} \right. \label{4.18}

Para un hamiltoniano independiente del tiempo es sencillo demostrar que los elementos de la matriz de densidad evolucionan como

\begin{align} \rho _ {n m} (t) &= \langle n | \rho (t) | m \rangle \\[4pt] &= \left\langle n | U | \psi _ {0} \right\rangle \left\langle \psi _ {0} \left| U^{\dagger} \right| m \right\rangle \label{4.19} \\[4pt] &= e^{- i \omega _ {n m} \left( t - t _ {0} \right)} \rho _ {n m} \left( t _ {0} \right) \label{4.20} \end{align}

De esto vemos que las poblaciones,$$\rho _ {m n} (t) = \rho _ {n m} \left( t _ {0} \right)$$, son invariantes en el tiempo, y las coherencias oscilan en la división de energía$$\omega _ {n m}$$.

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