Ehrenfest theorem

The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. It is

${\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\langle [A,H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle$

where A is some QM operator and $\langle A\rangle$ is its expectation value. Ehrenfest's theorem fits neatly into the Heisenberg picture of quantum mechanics.

Ehrenfest's theorem is closely related to Liouville's theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. In fact, it is a general rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in Classical mechanics by changing the commutator into a Poisson bracket and multiplying by $i\hbar$.

The theorem can be shown to follow from the Lindblad equation, a master equation for the time evolution of a mixed state.

Derivation

Suppose some system is presently in a quantum state $\Phi$. If we want to know the instantaneous time derivative of the expectation value of A, that is, by definition

${\frac {d}{dt}}\langle A\rangle ={\frac {d}{dt}}\int \Phi ^{*}A\Phi ~dx^{3}=\int \left({\frac {\partial \Phi ^{*}}{\partial t}}\right)A\Phi ~dx^{3}+\int \Phi ^{*}\left({\frac {\partial A}{\partial t}}\right)\Phi ~dx^{3}+\int \Phi ^{*}A\left({\frac {\partial \Phi }{\partial t}}\right)~dx^{3}$

$=\int \left({\frac {\partial \Phi ^{*}}{\partial t}}\right)A\Phi ~dx^{3}+\left\langle {\frac {\partial A}{\partial t}}\right\rangle +\int \Phi ^{*}A\left({\frac {\partial \Phi }{\partial t}}\right)~dx^{3},$

where we are integrating over all space. Often (but not always) the operator A is time independent, so that its derivative is zero and we can ignore the middle term. If we apply the Schrödinger equation, we find that

${\frac {\partial \Phi }{\partial t}}={\frac {1}{i\hbar }}H\Phi$

and

${\frac {\partial \Phi ^{*}}{\partial t}}={\frac {-1}{i\hbar }}\Phi ^{*}H^{*}={\frac {-1}{i\hbar }}\Phi ^{*}H.$^[1]

Notice $H=H^{*}$ because the Hamiltonian is hermitian. Placing this into the above equation we have

${\frac {d}{dt}}\langle A\rangle ={\frac {1}{i\hbar }}\int \Phi ^{*}(AH-HA)\Phi ~dx^{3}+\left\langle {\frac {\partial A}{\partial t}}\right\rangle ={\frac {1}{i\hbar }}\langle [A,H]\rangle +\left\langle {\frac {\partial A}{\partial t}}\right\rangle .$

General example

For the very general example of a massive particle moving in a potential, the Hamiltonian is simply

$H(x,p,t)={\frac {p^{2}}{2m}}+V(x,t)$

where x is just the location of the particle. Suppose we wanted to know the instantaneous change in momentum p. Using Ehrenfest's theorem, we have

${\frac {d}{dt}}\langle p\rangle ={\frac {1}{i\hbar }}\langle [p,H]\rangle +\left\langle {\frac {\partial p}{\partial t}}\right\rangle ={\frac {1}{i\hbar }}\langle [p,V(x,t)]\rangle$

since p commutes with itself and since when represented in coordinate space, the momentum operator $p=-i\hbar \nabla$ then ${\frac {\partial p}{\partial t}}=0$. Also

${\frac {d}{dt}}\langle p\rangle =\int \Phi ^{*}V(x,t)\nabla \Phi ~dx^{3}-\int \Phi ^{*}\nabla (V(x,t)\Phi )~dx^{3}.$

After applying a product rule, we have

${\frac {d}{dt}}\langle p\rangle =\langle -\nabla V(x,t)\rangle =\langle F\rangle ,$

but we recognize this as Newton's second law. This is an example of the correspondence principle, the result manifests as Newton's second law in the case of having so many particles that the net motion is given exactly by the expectation value of a single particle.

Notes

1. ^ In Bra-ket notation

${\frac {\partial }{\partial t}}\langle \phi |x\rangle ={\frac {-1}{i\hbar }}\langle \phi |{\hat {H}}|x\rangle ={\frac {-1}{i\hbar }}\langle \phi |x\rangle H={\frac {-1}{i\hbar }}\Phi ^{*}H$

where ${\hat {H}}$ is the Hamiltonian operator, and H is the Hamiltonian represented in coordinate space (as is the case in the derivation above). In other words, we applied the adjoint operation to the entire Schrödinger equation, which flipped the order of operations for H and $\Phi$.