Kubo formula

Consider a quantum system described by the (time independent) Hamiltonian ${\displaystyle H_{0}}$. The expectation value of a physical quantity at equilibrium temperature ${\displaystyle T}$, described by the operator ${\displaystyle {\hat {A}}}$, can be evaluated as:

${\displaystyle \left\langle {\hat {A}}\right\rangle ={1 \over Z_{0}}\operatorname {Tr} \,\left[{\hat {\rho _{0}}}{\hat {A}}\right]={1 \over Z_{0}}\sum _{n}\left\langle n\left|{\hat {A}}\right|n\right\rangle e^{-\beta E_{n}}}$,

where ${\displaystyle \beta =1/k_{\rm {B}}T}$ is the thermodynamic beta, ${\displaystyle {\hat {\rho }}_{0}}$ is density operator, given by

${\displaystyle {\hat {\rho _{0}}}=e^{-\beta {\hat {H}}_{0}}=\sum _{n}|n\rangle \langle n|e^{-\beta E_{n}}}$

and ${\displaystyle Z_{0}=\operatorname {Tr} \,\left[{\hat {\rho }}_{0}\right]}$ is the partition function.

Suppose now that just above some time ${\displaystyle t=t_{0}}$ an external perturbation is applied to the system. The perturbation is described by an additional time dependence in the Hamiltonian:

${\displaystyle {\hat {H}}(t)={\hat {H}}_{0}+{\hat {V}}(t)\theta (t-t_{0}),}$

where ${\displaystyle \theta (t)}$ is the Heaviside function (1 for positive times, 0 otherwise) and ${\displaystyle {\hat {V}}(t)}$ is hermitian and defined for all t, so that ${\displaystyle {\hat {H}}(t)}$ has for positive ${\displaystyle t-t_{0}}$ again a complete set of real eigenvalues ${\displaystyle E_{n}(t).}$ But these eigenvalues may change with time.

However, one can again find the time evolution of the density matrix ${\displaystyle {\hat {\rho }}(t)}$ rsp. of the partition function ${\displaystyle Z(t)=\operatorname {Tr} \,\left[{\hat {\rho }}(t)\right],}$ to evaluate the expectation value of

${\displaystyle \left\langle {\hat {A}}\right\rangle ={\frac {\operatorname {Tr} \,\left[\rho (t)\,{\hat {A}}\right]}{\operatorname {Tr} \,\left[{\hat {\rho }}(t)\right]}}.}$

The time dependence of the states ${\displaystyle |n(t)\rangle }$ is governed by the Schrödinger equation

${\displaystyle i\hbar {\frac {\partial }{\partial t}}|n(t)\rangle ={\hat {H}}(t)|n(t)\rangle ,}$

which thus determines everything, corresponding of course to the Schrödinger picture. But since ${\displaystyle {\hat {V}}(t)}$ is to be regarded as a small perturbation, it is convenient to now use instead the interaction picture representation, ${\displaystyle \left|{\hat {n}}(t)\right\rangle ,}$ in lowest nontrivial order. The time dependence in this representation is given by ${\displaystyle |n(t)\rangle =e^{-i{\hat {H}}_{0}t/\hbar }\left|{\hat {n}}(t)\right\rangle =e^{-i{\hat {H}}_{0}t/\hbar }{\hat {U}}(t,t_{0})\left|{\hat {n}}(t_{0})\right\rangle ,}$ where by definition for all t and ${\displaystyle t_{0}}$ it is: ${\displaystyle \left|{\hat {n}}(t_{0})\right\rangle =e^{i{\hat {H}}_{0}t_{0}/\hbar }|n(t_{0})\rangle }$

To linear order in ${\displaystyle {\hat {V}}(t)}$, we have

${\displaystyle {\hat {U}}(t,t_{0})=1-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'{\hat {V}}{\mathord {\left(t'\right)}}}$.

Thus one obtains the expectation value of ${\displaystyle {\hat {A}}(t)}$ up to linear order in the perturbation:

${\displaystyle \left\langle {\hat {A}}(t)\right\rangle =\left\langle {\hat {A}}\right\rangle _{0}-{\frac {i}{\hbar }}\int _{t_{0}}^{t}dt'{1 \over Z_{0}}\sum _{n}e^{-\beta E_{n}}\left\langle n(t_{0})\left|{\hat {A}}(t){\hat {V}}{\mathord {\left(t'\right)}}-{\hat {V}}{\mathord {\left(t'\right)}}{\hat {A}}(t)\right|n(t_{0})\right\rangle }$,

thus[3]

The brackets ${\displaystyle \langle \rangle _{0}}$ mean an equilibrium average with respect to the Hamiltonian ${\displaystyle H_{0}.}$ Therefore, although the result is of first order in the perturbation, it involves only the zeroth-order eigenfunctions, which is usually the case in perturbation theory and moves away all complications which otherwise might arise for ${\displaystyle t>t_{0}}$.

The above expression is true for any kind of operators. (see also Second quantization)[4]

• Green–Kubo relations