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5.2: Time-Evolution of the Density Matrix

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    107239
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    The equation of motion for the density matrix follows naturally from the definition of \(\rho\) and the time-dependent Schrödinger equation.

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

    Equation \ref{4.14} is the Liouville-Von Neumann equation. It is isomorphic to the Heisenberg equation of motion, since \(ρ\) is also an operator. The solution to Equation \ref{4.14} is

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

    This can be demonstrated by first integrating Equation \ref{4.14} to obtain

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

    If we expand Equation \ref{4.16} by iteratively substituting into itself, the expression is the same as when we substitute

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

    into Equation \ref{4.15} and collect terms by orders of \(H(\tau)\).

    Note that Equation \ref{4.15} and the cyclic invariance of the trace imply that the time-dependent expectation value of an operator can be calculated either by propagating the operator (Heisenberg) or the density matrix (Schrödinger or interaction picture):

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

    For a time-independent Hamiltonian it is straightforward to show that the density matrix elements evolve as

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

    From this we see that populations, \(\rho _ {m n} (t) = \rho _ {n m} \left( t _ {0} \right)\), are time-invariant, and coherences oscillate at the energy splitting \(\omega _ {n m}\).


    5.2: Time-Evolution of the Density Matrix is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Andrei Tokmakoff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.