The Kubo formula, named for Ryogo Kubo who first presented the formula in 1957,[1] [2] is an equation which expresses the linear response of an observable quantity due to a time-dependent perturbation.
Among numerous applications of the Kubo formula, one can calculate the charge and spin susceptibilities of systems of electrons in response to applied electric and magnetic fields. Responses to external mechanical forces and vibrations can be calculated as well.
Consider a quantum system described by the (time independent) Hamiltonian
H0
T
\hat{A}
\left\langle\hat{A}\right\rangle={1\overZ0}\operatorname{Tr}\left[\hat{\rho0}\hat{A}\right]={1\overZ0}\sumn\left\langlen\left|\hat{A}\right|n\right\rangle
-\betaEn | |
e |
\beta=1/k\rmT
\hat{\rho}0
\hat{\rho0}=
-\beta\hat{H | |
e | |
0} |
=\sumn|n\rangle\langlen|
-\betaEn | |
e |
Z0=\operatorname{Tr}\left[\hat\rho0\right]
Suppose now that just above some time
t=t0
\hat{H}(t)=\hat{H}0+\hat{V}(t)\theta(t-t0),
\theta(t)
\hatV(t)
\hatH(t)
t-t0
En(t).
\hat{\rho}(t)
Z(t)=\operatorname{Tr}\left[\hat\rho(t)\right],
\left\langle\hatA\right\rangle=
\operatorname{Tr | |
\left[\hat |
\rho(t)\hatA\right]}{\operatorname{Tr}\left[\hat\rho(t)\right]}.
The time dependence of the states
i\hbar | \partial |
\partialt |
|n(t)\rangle=\hat{H}(t)|n(t)\rangle,
\hat{V}(t)
\left|\hatn(t)\right\rangle,
|n(t)\rangle=
-i\hatH0t/\hbar | |
e |
\left|\hat{n}(t)\right\rangle=
-i\hatH0t/\hbar | |
e |
\hat{U}(t,t0)\left|\hat{n}(t0)\right\rangle,
t0
\left|\hat{n}(t0)\right\rangle=
i\hatH0t0/\hbar | |
e |
|n(t0)\rangle
To linear order in
\hat{V}(t)
\hat{U}(t,t0)=1-
i | |
\hbar |
t | |
\int | |
t0 |
dt'\hat{V}d\left(t'\right)
\hat{A}(t)
\left\langle\hat{A}(t)\right\rangle=\left\langle\hat{A}\right\rangle0-
i | |
\hbar |
t | |
\int | |
t0 |
dt'{1\overZ0}\sumn
-\betaEn | |
e |
\left\langlen(t0)\left|\hat{A}(t)\hat{V}d\left(t'\right)-\hat{V}d\left(t'\right)\hat{A}(t)\right|n(t0)\right\rangle
The brackets
\langle\rangle0
H0.
t>t0
The above expression is true for any kind of operators. (see also Second quantization)[4]