# Time Evolution of Expectation Values and Ehrenfest's Theorem

## Time Evolution of Expecation Values

Having described in a 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 $\hat{O}(t),$ we know that its expectation value is given by $\langle\hat{O}(t)\rangle=\langle\Psi(t)|\hat{O}(t)|\Psi(t)\rangle.$ If we take the time derivative of this expectation value, we get

$\frac{d\langle\hat{O}(t)\rangle}{dt}=\left [\frac{d}{dt}\langle\Psi(t)|\right ]\hat{O}(t)|\Psi(t)\rangle+\langle\Psi(t)|\frac{d\hat{O}(t)}{dt}|\Psi(t)\rangle+\langle\Psi(t)|\hat{O}(t)\left [\frac{d}{dt}|\Psi(t)\rangle\right ]$

$=\left [\frac{d}{dt}\langle\Psi(t)|\right ]\hat{O}(t)|\Psi(t)\rangle+\langle\Psi(t)|\hat{O}(t)\left [\frac{d}{dt}|\Psi(t)\rangle\right ]+\left\langle\frac{d\hat{O}(t)}{dt}\right\rangle.$

We now use the Schrödinger equation and its dual to write this as

$\frac{d\langle\hat{O}(t)\rangle}{dt}=\frac{i}{\hbar}\langle\Psi(t)|\hat{H}(t)\hat{O}(t)|\Psi(t)\rangle-\frac{i}{\hbar}\langle\Psi(t)|\hat{O}(t)\hat{H}(t)|\Psi(t)\rangle+\left\langle\frac{d\hat{O}(t)}{dt}\right\rangle$

$=\frac{i}{\hbar}\langle[\hat{H}(t),\hat{O}(t)]\rangle+\left\langle\frac{d\hat{O}(t)}{dt}\right\rangle.$

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, because the expectation values, in the classical limit, become the values of the corresponding classical quantities.

Consider the Hamiltonian,

$\hat{H}=\frac{\hat{\mathbf{p}}^2}{2m}+V(\hat{\mathbf{r}}).$

We are now interested in determining how the expecation values of the position $\hat{\mathbf{r}}$ and momentum $\hat{\mathbf{p}}$ operators evolve in time. Using the formula that we just derived, and noting that neither operator depends explicitly on time, we obtain

$\frac{d\langle\hat{\mathbf{r}}\rangle}{dt}=\frac{i}{\hbar}\langle[\hat{H},\hat{\mathbf{r}}]\rangle$ and $\frac{d\langle\hat{\mathbf{p}}\rangle}{dt}=\frac{i}{\hbar}\langle[\hat{H},\hat{\mathbf{p}}]\rangle.$

Using the fact that $[\hat{p}_i^2,\hat{x}_j]=-2i\hbar p_i\delta_{ij}$ and $[p_i,f(\hat{\mathbf{r}})]=-i\hbar\frac{\partial f(\hat{\mathbf{r}})}{\partial\hat{x}_i},$ we find that

$\frac{d\langle\hat{\mathbf{r}}\rangle}{dt}=\frac{\langle\hat{\mathbf{p}}\rangle}{m}$ and $\frac{d\langle\hat{\mathbf{p}}\rangle}{dt}=-\langle\nabla V(\hat{\mathbf{r}})\rangle.$

These two equations closely resemble equations familar from classical mechanics - the first resembles the statement that momentum is equal to mass times velocity, while the latter looks like Newton's second law.