[NOTES/QM-11007] Time Variation of Average Values

The time evolution of a quantum system is governed by the Schrodinger equation \begin{equation} i \hbar \dd{t}\psi(x, t)=\hat{H}\psi(x, t). \end{equation} We will obtain an equation for time development of averages of a dynamical variable $\hat{F}$. The result will turn out to have an obvious correspondence with the classical equation of motion for dynamical variable $F$. We will take the Hamiltonian to be the operator for a single particle in coordinate representation \begin{equation}\label{eq10} H = -\frac{\hbar^2}{2m} \DD{x} + V(x) \end{equation} Let $F(q,p,t)$ be an dynamical variable of the system and let $\hat{F}$ denote the corresponding operator. We are interested in finding out how the average value \begin{equation} \langle \hat{F} \rangle_\psi \equiv \int dx\, \psi(x, t)\hat{F} \psi(x, t) \label{E13} \end{equation} changes with time. The time dependence of the average value comes from dependence of the three objects, the operator$\hat{F}$, and the wave function appearing as \(\psi(x,t), \psi^*(x,t)\) in \eqref{E13}. The equation conjugate to the Schrodinger equation \begin{equation}\label{eq03} i\hbar {d\over d t}\,\psi(x, t) = \Big[ -\frac{\hbar^2}{2m} \DD{x} + V(x)\Big] \psi(x,t) \end{equation} is given by \begin{equation} - i\hbar {d\over d t}\, \psi^*(x,t) =\Big[ -\frac{\hbar^2}{2m} \DD{x} + V(x) \Big] \psi(x, t) \label{E14} \end{equation} Therefore \begin{eqnarray}\nonumber \pp{t}\langle \hat{F} \rangle_\psi = \int dx\, \left(\frac{d}{dt}\psi^*(x,t)\right)\hat{F}\psi(x,t) &+& \int dx\,\psi(x, t)\Big(\frac{d\hat{F}}{dt}\Big)\psi(x,t) \\ & + &\int dx\, \psi^*(x,t)\hat{F} \bigg(\pp{t} \psi(x,t)\bigg) \label{E16} \end{eqnarray} Using \eqref{eq03} and \eqref{E14} in \eqref{E16} we get \begin{eqnarray}\nonumber {d\over dt}\langle \hat{F} \rangle_\psi = - {1\over i \hbar}\int dx\,\big(H \psi(x,t)\big)^* \hat{F}\psi(x,t) &+& \int dx\,\psi^*(x,t){d\hat{F} \over dt} \psi(x, t)\\ &+&{1\over i \hbar}\int dx\, \psi^*(x,t)\hat{F} \hat{H} \psi(x,t). \label{E17} \end{eqnarray} Using the hermiticity property of the Hamiltonian operator, the above equation can now be rearranged to give the final form
\begin{equation} {d\over dt}\, \langle \hat{F} \rangle = \,\langle{\partial\over \partial t} \hat{F} \rangle + {1\over i\hbar} \langle\, [\hat{F},\hat{H} ]\, \rangle. \label{E18} \end{equation}
This result is known as Ehrenfest theorem. Comparing \eqref{E18} with the equation of motion in classical mechanics for time evolution of dynamical variables
\begin{equation} {d F\over d t} = {\partial F\over \partial t} + \{F,H\}_{PB} \label{E19} \end{equation}
and remembering that the commutator divided by $i\hbar$ corresponds to the Poisson bracket in the limit $\hbar \rightarrow 0$, we see that $\hat{H}$ must be identified as the operator corresponding to the Hamiltonian $H$ of the system. For a particle in potential \(V(x)\) we have \begin{eqnarray} \dd{t} \langle p \rangle &=& -\Big\langle\dd[V(x)]{x} \Big\rangle\\ \dd{t}\langle x \rangle &=& \Big\langle \frac{p}{m} \Big\rangle \end{eqnarray} Thus we have the result that the rate of change of average values obey equations similar to those in classical theory. This result is known as Ehrenfest theorem. This, for example, implies that the center of mass of the quantum state will approximately follow a classical trajectory.