---
title: "Ehrenfest's theorem"
date: 2021-02-24
categories:
- Quantum mechanics
- Physics
layout: "concept"
---

In quantum mechanics, **Ehrenfest's theorem** gives a general expression for the
time evolution of an observable's expectation value $\expval{\hat{L}}$.

The time-dependent Schrödinger equation is as follows,
where prime denotes differentiation with respect to time $t$:

$$\begin{aligned}
    \Ket{\psi'} = \frac{1}{i \hbar} \hat{H} \Ket{\psi}
    \qquad
    \Bra{\psi'} = - \frac{1}{i \hbar} \Bra{\psi} \hat{H}
\end{aligned}$$

Given an observable operator $\hat{L}$ and a state $\Ket{\psi}$,
the time-derivative of the expectation value $\expval{\hat{L}}$ is as follows
(due to the product rule of differentiation):

$$\begin{aligned}
    \dv{\expval{\hat{L}}}{t}
    &= \matrixel{\psi}{\hat{L}}{\psi'} + \matrixel{\psi'}{\hat{L}}{\psi} + \matrixel{\psi}{\hat{L}'}{\psi}
    \\
    &= \frac{1}{i \hbar} \matrixel{\psi}{\hat{L}\hat{H}}{\psi}
    - \frac{1}{i \hbar} \matrixel{\psi}{\hat{H}\hat{L}}{\psi}
    + \Expval{\dv{\hat{L}}{t}}
\end{aligned}$$

The first two terms on the right can be rewritten using a commutator,
yielding the general form of Ehrenfest's theorem:

$$\begin{aligned}
    \boxed{
        \dv{\expval{\hat{L}}}{t}
        = \frac{1}{i \hbar} \Expval{[\hat{L}, \hat{H}]} + \Expval{\dv{\hat{L}}{t}}
    }
\end{aligned}$$

In practice, since most operators are time-independent,
the last term often vanishes.

As a interesting side note, in the [Heisenberg picture](/know/concept/heisenberg-picture/),
this relation proves itself,
when one simply wraps all terms in $\Bra{\psi}$ and $\Ket{\psi}$.

Two observables of particular interest are the position $\hat{X}$ and momentum $\hat{P}$.
Applying the above theorem to $\hat{X}$ yields the following,
which we reduce using the fact that $\hat{X}$ commutes
with the potential $V(\hat{X})$,
because one is a function of the other:

$$\begin{aligned}
    \dv{\expval{\hat{X}}}{t}
    &= \frac{1}{i \hbar} \Expval{[\hat{X}, \hat{H}]}
    = \frac{1}{2 i \hbar m} \Expval{[\hat{X}, \hat{P}^2] + 2 m [\hat{X}, V(\hat{X})]}
    = \frac{1}{2 i \hbar m} \Expval{[\hat{X}, \hat{P}^2]}
    \\
    &= \frac{1}{2 i \hbar m} \Expval{\hat{P} [\hat{X}, \hat{P}] + [\hat{X}, \hat{P}] \hat{P}}
    = \frac{2 i \hbar}{2 i \hbar m} \expval{\hat{P}}
    = \frac{\expval{\hat{P}}}{m}
\end{aligned}$$

This is the first part of the "original" form of Ehrenfest's theorem,
which is reminiscent of classical Newtonian mechanics:

$$\begin{gathered}
    \boxed{
        \dv{\expval{\hat{X}}}{t} = \frac{\expval{\hat{P}}}{m}
    }
\end{gathered}$$

Next, applying the general formula to the expected momentum $\expval{\hat{P}}$
gives us:

$$\begin{aligned}
    \dv{\expval{\hat{P}}}{t}
    &= \frac{1}{i \hbar} \Expval{[\hat{P}, \hat{H}]}
    = \frac{1}{2 i \hbar m} \Expval{[\hat{P}, \hat{P}^2] + 2 m [\hat{P}, V(\hat{X})]}
    = \frac{1}{i \hbar} \Expval{[\hat{P}, V(\hat{X})]}
\end{aligned}$$

To find the commutator, we go to the $\hat{X}$-basis and use a test
function $f(x)$:

$$\begin{aligned}
    \Comm{- i \hbar \dv{}{x}}{V(x)} \: f(x)
    &= - i \hbar \frac{dV}{dx} f(x) - i \hbar V(x) \frac{df}{dx} + i \hbar V(x) \frac{df}{dx}
    = - i \hbar \frac{dV}{dx} f(x)
\end{aligned}$$

By inserting this result back into the previous equation, we find the following:

$$\begin{aligned}
    \dv{\expval{\hat{P}}}{t}
    &= - \frac{i \hbar}{i \hbar} \Expval{\frac{d V}{d \hat{X}}}
    = - \Expval{\frac{d V}{d \hat{X}}}
\end{aligned}$$

This is the second part of Ehrenfest's theorem,
which is also similar to Newtonian mechanics:

$$\begin{gathered}
    \boxed{
        \dv{\expval{\hat{P}}}{t} = - \Expval{\pdv{V}{\hat{X}}}
    }
\end{gathered}$$

There is an important consequence of Ehrenfest's original theorems
for the symbolic derivatives of the Hamiltonian $\hat{H}$
with respect to $\hat{X}$ and $\hat{P}$:

$$\begin{gathered}
    \boxed{
        \Expval{\pdv{\hat{H}}{\hat{P}}}
        = \dv{\expval{\hat{X}}}{t}
    }
    \qquad \quad
    \boxed{
        - \Expval{\pdv{\hat{H}}{\hat{X}}}
        = \dv{\expval{\hat{P}}}{t}
    }
\end{gathered}$$

These are easy to prove yourself,
and are analogous to Hamilton's canonical equations.