Time Evolution of Expectation Values and Ehrenfest's Theorem
Time Evolution of Expecation Values
Having described in the previous section how the state vector of a system evolves in time, we may now derive a formula for the time evolution of the expectation value of an operator. Given an operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{O}(t),} we know that its expectation value is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat{O}(t)\rangle=\langle\Psi(t)|\hat{O}(t)|\Psi(t)\rangle.} If we take the time derivative of this expectation value, we get
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\langle\hat{O}(t)\rangle}{dt}=\left [\frac{d}{dt}\langle\Psi(t)|\right ]\hat{O}(t)|\Psi(t)\rangle+\langle\Psi(t)|\frac{d\hat{O}(t)}{dt}|\Psi(t)\rangle+\langle\Psi(t)|\hat{O}(t)\left [\frac{d}{dt}|\Psi(t)\rangle\right ]}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\left [\frac{d}{dt}\langle\Psi(t)|\right ]\hat{O}(t)|\Psi(t)\rangle+\langle\Psi(t)|\hat{O}(t)\left [\frac{d}{dt}|\Psi(t)\rangle\right ]+\left\langle\frac{d\hat{O}(t)}{dt}\right\rangle.}
We now use the Schrödinger equation and its dual to write this as
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\langle\hat{O}(t)\rangle}{dt}=\frac{i}{\hbar}\langle\Psi(t)|\hat{H}(t)\hat{O}(t)|\Psi(t)\rangle-\frac{i}{\hbar}\langle\Psi(t)|\hat{O}(t)\hat{H}(t)|\Psi(t)\rangle+\left\langle\frac{d\hat{O}(t)}{dt}\right\rangle}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{i}{\hbar}\langle[\hat{H}(t),\hat{O}(t)]\rangle+\left\langle\frac{d\hat{O}(t)}{dt}\right\rangle.}
This formula is of the utmost importance in all facets of quantum mechanics.
Ehrenfest's Theorem
We now use the above result to prove Ehrenfest's Theorem, which states that the expecation values of the position and momentum operators obey the same equations that the corresponding classical quantities obey. Thus, one may consider this theorem to be a manifestation of the correspondence principle.
Consider the Hamiltonian,
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{H}=\frac{\hat{\mathbf{p}}^2}{2m}+V(\hat{\mathbf{r}}).}
We are now interested in determining how the expecation values of the position Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{r}}} and momentum Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{p}}} operators evolve in time. Using the formula that we just derived, and noting that neither operator depends explicitly on time,