5.2: Evolución temporal de la matriz de densidad
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- 73789
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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\(\omega _ {n m}\).